{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0001","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3347ce5ad58b2eb15c414b64e9ee911cfcb5cc977168cfcb113860421ad23baf","family_id":"qf-rate-endpoint-quotient","seed":202610063001,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"1\",\"0\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (-13, -9) and (-12, -9), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"0/1","explanation":"The output difference is 0 and the input difference is 1, giving 0/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0002","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5d784f7c555c8a114cdeb75a03081ce7ff92227879a1af03a0547903c2943b0","family_id":"qf-rate-endpoint-quotient","seed":202610063002,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"2\",\"-2\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (-12, -7) and (-14, -5), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-1/1","explanation":"The output difference is 2 and the input difference is -2, giving -1/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0003","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"016501dd67674b52a509eb74d555dfa1b8099e010955e233b26b77356454ce82","family_id":"qf-rate-endpoint-quotient","seed":202610063003,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"3\",\"3\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (-11, -5) and (-8, -2), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"1/1","explanation":"The output difference is 3 and the input difference is 3, giving 1/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0004","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21ed7b51d54a13e96d7c1b19cd9f8ea04aaec66ddcba5ea9ae48c72dad57f43f","family_id":"qf-rate-endpoint-quotient","seed":202610063004,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"4\",\"-4\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (-10, -3) and (-14, 1), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-1/1","explanation":"The output difference is 4 and the input difference is -4, giving -1/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0005","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42794775678ae2d2160630ca07d2894a98713069b07de952eeba3d931286ce96","family_id":"qf-rate-endpoint-quotient","seed":202610063005,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"5\",\"5\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (-9, -1) and (-4, 4), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"1/1","explanation":"The output difference is 5 and the input difference is 5, giving 1/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0006","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"891e3c370a5c3797206a42da50739d541c20d1fd10d268134578936b1495a950","family_id":"qf-rate-endpoint-quotient","seed":202610063006,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"6\",\"-6\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (-8, 1) and (-14, 7), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-1/1","explanation":"The output difference is 6 and the input difference is -6, giving -1/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0007","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"157ea48db09b1910fa80742b571eedacffb92de64e2e2f3d597e1730c4793890","family_id":"qf-rate-endpoint-quotient","seed":202610063007,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"7\",\"7\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (-7, 3) and (0, 10), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"1/1","explanation":"The output difference is 7 and the input difference is 7, giving 1/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0008","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e1ff364c07ff2031e26adeec5c6776578a6ed149d97f6b06c5af4b8cf117272","family_id":"qf-rate-endpoint-quotient","seed":202610063008,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"1\",\"-8\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (-6, 5) and (-7, 13), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-8/1","explanation":"The output difference is 8 and the input difference is -1, giving -8/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0009","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc40921cc0009680974a666e78a8614c7985c1cd8bc1e545f9ac009bbd0235fb","family_id":"qf-rate-endpoint-quotient","seed":202610063009,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"2\",\"9\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (-5, 7) and (-3, 16), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"9/2","explanation":"The output difference is 9 and the input difference is 2, giving 9/2. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0010","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1af73fed422fd98d38c5a3ca06913ee96d114f354da27241af5cbdee938e6c4","family_id":"qf-rate-endpoint-quotient","seed":202610063010,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"3\",\"-10\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (-4, 9) and (-7, 19), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-10/3","explanation":"The output difference is 10 and the input difference is -3, giving -10/3. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0011","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cb4d5a25782449342872d813c338ff255bdb57775f4c04e57bc45d0e0b1b4ac","family_id":"qf-rate-endpoint-quotient","seed":202610063011,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"4\",\"11\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (-3, 11) and (1, 22), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"11/4","explanation":"The output difference is 11 and the input difference is 4, giving 11/4. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0012","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7531a505ebb4375867728d4d75778c2a496e45db3d0428e79ac9941a1cde077","family_id":"qf-rate-endpoint-quotient","seed":202610063012,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"5\",\"-12\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (-2, 13) and (-7, 25), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-12/5","explanation":"The output difference is 12 and the input difference is -5, giving -12/5. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0013","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48d8f482f8c9a13964a2416fac5eab561fb2edfbc1bc64e11925112548eafafa","family_id":"qf-rate-endpoint-quotient","seed":202610063013,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"6\",\"13\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (-1, 15) and (5, 28), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"13/6","explanation":"The output difference is 13 and the input difference is 6, giving 13/6. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0014","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32c7da6d35ca40c86c98653dc34686b8906e4379ab5ffbb954e2910d8ae06200","family_id":"qf-rate-endpoint-quotient","seed":202610063014,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"7\",\"-14\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (0, 17) and (-7, 31), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-2/1","explanation":"The output difference is 14 and the input difference is -7, giving -2/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0015","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e08fdd10024f07e17f811009f0b8e4ac5e6d1af497ed49a05957d15acfa1b104","family_id":"qf-rate-endpoint-quotient","seed":202610063015,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"1\",\"15\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (1, 19) and (2, 34), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"15/1","explanation":"The output difference is 15 and the input difference is 1, giving 15/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0016","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"536a36082f84ba344af9e06f567538a77a682f70ca43537cfc6cad3b8c8e6a4e","family_id":"qf-rate-endpoint-quotient","seed":202610063016,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"2\",\"-16\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (2, 21) and (0, 37), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-8/1","explanation":"The output difference is 16 and the input difference is -2, giving -8/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0017","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35f1780efdff59c76a69558ec1cfb15377736dda8ba6ba866f782f7ccbfdf7b7","family_id":"qf-rate-endpoint-quotient","seed":202610063017,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"3\",\"17\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (3, 23) and (6, 40), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"17/3","explanation":"The output difference is 17 and the input difference is 3, giving 17/3. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0018","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"841a79a17c385dc3815ffa473084afe949ffbcad29e4237cb88b54a4a99ac4c3","family_id":"qf-rate-endpoint-quotient","seed":202610063018,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"4\",\"-18\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (4, 25) and (0, 43), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-9/2","explanation":"The output difference is 18 and the input difference is -4, giving -9/2. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0019","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f42ae951b22296d28dab0ede2002e1181ba6591d1865c927b0150c0facb929b","family_id":"qf-rate-endpoint-quotient","seed":202610063019,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"5\",\"19\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (5, 27) and (10, 46), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"19/5","explanation":"The output difference is 19 and the input difference is 5, giving 19/5. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0020","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"790fcc54ef8b156748a23bb84f99dcfb7dac68219d1078dccb60d5fc64ddb9e9","family_id":"qf-rate-endpoint-quotient","seed":202610063020,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"6\",\"-20\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (6, 29) and (0, 49), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-10/3","explanation":"The output difference is 20 and the input difference is -6, giving -10/3. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0021","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7111acf66f94e7e37c2f10a04fe9b2815210357a7c462088052eb9ae319b49fe","family_id":"qf-rate-endpoint-quotient","seed":202610063021,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"7\",\"21\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (7, 31) and (14, 52), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"3/1","explanation":"The output difference is 21 and the input difference is 7, giving 3/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0022","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2223fd54df96b3fb73558fceff86cae729cf41bad29280bc4073a8b5af64c502","family_id":"qf-rate-endpoint-quotient","seed":202610063022,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"1\",\"-22\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (8, 33) and (7, 55), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-22/1","explanation":"The output difference is 22 and the input difference is -1, giving -22/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0023","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc45bace395b8062475ea67215ace742d40ef0fe9fc83c568a8da86528449a27","family_id":"qf-rate-endpoint-quotient","seed":202610063023,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"2\",\"23\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (9, 35) and (11, 58), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"23/2","explanation":"The output difference is 23 and the input difference is 2, giving 23/2. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0024","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5878b85fa29112f99b4af22738394f5e8b571c38e2c7360a34253f4f793db6fd","family_id":"qf-rate-endpoint-quotient","seed":202610063024,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"3\",\"-24\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (10, 37) and (7, 61), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-8/1","explanation":"The output difference is 24 and the input difference is -3, giving -8/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0025","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c01457f0a12c4c3ca01222fd97887c5a2f803e65656f53da66f8a48883e99e38","family_id":"qf-rate-endpoint-quotient","seed":202610063025,"semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"4\",\"25\"],\"operation\":\"endpoint_quotient\"}"},"fields":{"title":"Average Rate between Explicit Function Endpoints","stem":"For the same specified function, the permitted endpoints are (11, 39) and (15, 64), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"25/4","explanation":"The output difference is 25 and the input difference is 4, giving 25/4. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative."},"skills":["Average Rate between Explicit Function Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0026","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9753bd843f30963f62bf4fee1c2b5051c7a126078613e547976b6e723fc5da2","family_id":"qf-rate-denominator-validity","seed":202610063026,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"0\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=0 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"No","explanation":"The input difference is 0, so the eligibility answer is No. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0027","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fa5f509e34815a019c8fca684342bf84d61a7db130a49874711713060a4b20d","family_id":"qf-rate-denominator-validity","seed":202610063027,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"1\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-1 and x1=-2. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is -1, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0028","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9e3822517cdb60454b66f5c3a5c52b546b62b4f52d9ff2645fcba29ac850cd2","family_id":"qf-rate-denominator-validity","seed":202610063028,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"2\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-2 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is 2, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0029","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c412f73ac6a3233144b2a0b7efc0bdd2f9a0a282244b1052b223e572dd70aef","family_id":"qf-rate-denominator-validity","seed":202610063029,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"3\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-3 and x1=-6. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is -3, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0030","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd317e8be306f84fb6b59e45f72522b1205c4357b583adc22fcbb9c70129fbec","family_id":"qf-rate-denominator-validity","seed":202610063030,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"4\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-4 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is 4, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0031","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b04ddfcbe19c0afd4db4e45002e162e949490a1a71239a57657def8ed2ca850f","family_id":"qf-rate-denominator-validity","seed":202610063031,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"5\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-5 and x1=-10. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is -5, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0032","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff77ce4ab8aee7a138d193bfd95c0949f31b979c5dcd5d34fba782b9b784b94a","family_id":"qf-rate-denominator-validity","seed":202610063032,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"6\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-6 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is 6, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0033","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6555e410ab9ac83d919f95175a0730e7dcc3bd06a2f69f1f469d6d42f8a77d6a","family_id":"qf-rate-denominator-validity","seed":202610063033,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"7\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-7 and x1=-14. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is -7, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0034","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a62ebc3710a89d68d0afcac18a33676c6818dea2e7c688e920e601fb643b578","family_id":"qf-rate-denominator-validity","seed":202610063034,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"8\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-8 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is 8, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0035","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6c459bc4be222898ad6064e730132b0a8b671e6d5c2d0dae1ee4ba7933c1ea1","family_id":"qf-rate-denominator-validity","seed":202610063035,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"9\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-9 and x1=-18. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is -9, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0036","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1756c01980e8c7a2ea655c3819f9f3a4b51b5c8d1261461c31404259cdf390ba","family_id":"qf-rate-denominator-validity","seed":202610063036,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"10\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-10 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is 10, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0037","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5aca378afd300f2b2ca16b650f4d489a97c59956887867f3756c0ebdbbb5db55","family_id":"qf-rate-denominator-validity","seed":202610063037,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"11\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-11 and x1=-22. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is -11, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0038","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03fd332a2b6ad984013c12ba6a1bbd625a60a0e0f213d2a818c6c5137e726927","family_id":"qf-rate-denominator-validity","seed":202610063038,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"12\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-12 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is 12, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0039","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3167ca9121598f299c3a5a275ffb99de3fba096bc27e4251231b332c5e148089","family_id":"qf-rate-denominator-validity","seed":202610063039,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"13\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-13 and x1=-26. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is -13, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0040","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"772b0d09bd1edcb1bbb125db6d3a9009e50acea84d22b89e5cfbe398530064f0","family_id":"qf-rate-denominator-validity","seed":202610063040,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"14\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-14 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is 14, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0041","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08268e7f9aa992dbea425e410b9bffba581f09e4ac7f0e7aa8f41bf84e96bd86","family_id":"qf-rate-denominator-validity","seed":202610063041,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"15\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-15 and x1=-30. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is -15, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0042","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27e3cebe8ef285c470b8141cc34310da14b504015eead95890a93d03a3524d07","family_id":"qf-rate-denominator-validity","seed":202610063042,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"16\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-16 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is 16, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0043","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f74ca065d4ceff1e2c44f75887a6f22080d0a4cbd6473af83771ea572bf68c98","family_id":"qf-rate-denominator-validity","seed":202610063043,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"17\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-17 and x1=-34. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is -17, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0044","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f81738dc2695298a7effa06d0813f4caabbbf1bdfd576cfb4207d3ff591b4bec","family_id":"qf-rate-denominator-validity","seed":202610063044,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"18\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-18 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is 18, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0045","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ba4c9eb909934d61e5f4feefe418d694b68e2375b75714670d769d6d956d88a","family_id":"qf-rate-denominator-validity","seed":202610063045,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"19\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-19 and x1=-38. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is -19, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0046","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f07e42d53175e6fe87565d926764b1706ba060b1b7a84fd7aca70ee5710182a","family_id":"qf-rate-denominator-validity","seed":202610063046,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"20\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-20 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is 20, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0047","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2e7596f70eafad718c42bc749106ddf8a8a7453bc97843cdf8cb34ab1ca3af7","family_id":"qf-rate-denominator-validity","seed":202610063047,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"21\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-21 and x1=-42. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is -21, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0048","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5bcd5d4043cc91009eb4a048cb7d7cd52a7a109be49b1e5c78b83c1a4cf6938","family_id":"qf-rate-denominator-validity","seed":202610063048,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"22\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-22 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is 22, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0049","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfbe09ad0847aad33b42962835c3361c4aaeb7cfd78fe8948be07d7167c197e8","family_id":"qf-rate-denominator-validity","seed":202610063049,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"23\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-23 and x1=-46. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is -23, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0050","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de1debf2eb8ac0dbe1642096e93ae51897b31c540f68a87fe0662ab1c228c14e","family_id":"qf-rate-denominator-validity","seed":202610063050,"semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"24\"],\"operation\":\"denominator_validity\"}"},"fields":{"title":"Nonzero-Denominator Eligibility for an Average Rate","stem":"The proposed permitted endpoint inputs are x0=-24 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?","answer":{"kind":"short_text","value":"Yes","explanation":"The input difference is 24, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative."},"skills":["Nonzero-Denominator Eligibility for an Average Rate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0051","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8028acc40641cce63f9ce459e9eb2bd92eba827211647b8cf25e8d0f63fac3dd","family_id":"qf-rate-affine-average","seed":202610063051,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-12\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(-12)x+(-7). For distinct inputs 0 and 1, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-12/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=-12. The exact average rate is -12/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0052","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dfcc398d81c90afce8fb4fb93c7ffc4b5cb6276daeed5655e32ea098c9acf51","family_id":"qf-rate-affine-average","seed":202610063052,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-11\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(-11)x+(-4). For distinct inputs 0 and -2, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-11/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=-11. The exact average rate is -11/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0053","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1416439adc973b0d46c5dfaea5c026d691dcd483f871aca367999bb7611a979d","family_id":"qf-rate-affine-average","seed":202610063053,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-10\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(-10)x+(-1). For distinct inputs 0 and 3, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-10/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=-10. The exact average rate is -10/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0054","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abb523b37c9096096a34ab1c7c4fef1607f0229f2cc069b47dd1b7891100a17e","family_id":"qf-rate-affine-average","seed":202610063054,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-9\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(-9)x+(2). For distinct inputs 0 and -4, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-9/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=-9. The exact average rate is -9/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0055","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acc162758bcc62ae6a0a74504c91fc207bcf20aff7a148bd861365364a51053d","family_id":"qf-rate-affine-average","seed":202610063055,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-8\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(-8)x+(5). For distinct inputs 0 and 5, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-8/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=-8. The exact average rate is -8/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0056","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49921a25da0baf5a1e8d3760ea175b5a74df8904a41a12d82abdc150dc7e14a4","family_id":"qf-rate-affine-average","seed":202610063056,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-7\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(-7)x+(8). For distinct inputs 0 and -6, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-7/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=-7. The exact average rate is -7/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0057","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba98c0d89973a06ad86871ef94f836dd17c0cb49386537e3fa1215e8788f2581","family_id":"qf-rate-affine-average","seed":202610063057,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-6\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(-6)x+(11). For distinct inputs 0 and 7, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-6/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=-6. The exact average rate is -6/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0058","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46e03204d76db850f1c1784585b63d75719c3cf2306fd049403decfd8813f40b","family_id":"qf-rate-affine-average","seed":202610063058,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-5\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(-5)x+(14). For distinct inputs 0 and -8, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-5/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=-5. The exact average rate is -5/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0059","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee278dcf3b935fb291e99fefa03dc07ec50e2b74f49976d29bed881f7490ef74","family_id":"qf-rate-affine-average","seed":202610063059,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-4\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(-4)x+(17). For distinct inputs 0 and 9, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-4/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=-4. The exact average rate is -4/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0060","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce784902f48e4118f3ebd70d5afa777abb91cf94f33bbec2958c36b7986776c3","family_id":"qf-rate-affine-average","seed":202610063060,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-3\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(-3)x+(20). For distinct inputs 0 and -10, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-3/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=-3. The exact average rate is -3/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0061","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"737e860fca523ed2921f396f3c2b810fb8597a974b13e6a01283a134d5f6b569","family_id":"qf-rate-affine-average","seed":202610063061,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-2\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(-2)x+(23). For distinct inputs 0 and 11, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-2/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=-2. The exact average rate is -2/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0062","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b38181913aeb465d85e1acf05c90b9fc1c68c7f52746fc2c2c4ced8eb074fbc","family_id":"qf-rate-affine-average","seed":202610063062,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-1\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(-1)x+(26). For distinct inputs 0 and -12, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-1/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=-1. The exact average rate is -1/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0063","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af70e868526abe6ca27aa08873332e1c2c38cfcdbca4135c2d56b9b0560431fd","family_id":"qf-rate-affine-average","seed":202610063063,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"0\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(0)x+(29). For distinct inputs 0 and 13, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"0/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=0. The exact average rate is 0/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0064","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"134a236b61b6b527e628b5ad3e28435ef876f07ec71a7483d37f283a6806821c","family_id":"qf-rate-affine-average","seed":202610063064,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"1\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(1)x+(32). For distinct inputs 0 and -14, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"1/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=1. The exact average rate is 1/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0065","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5267ae65d139f530a8a0e09a421004d9feeee087548a3f55945f850de54e29e6","family_id":"qf-rate-affine-average","seed":202610063065,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"2\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(2)x+(35). For distinct inputs 0 and 15, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"2/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=2. The exact average rate is 2/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0066","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d3f1ecea2cfca8c55d6624a1bbbf82c9c1558f1f4e87c62c5bfcccb556b58ce","family_id":"qf-rate-affine-average","seed":202610063066,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"3\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(3)x+(38). For distinct inputs 0 and -16, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"3/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=3. The exact average rate is 3/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0067","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adbd7bc45004547f86f80952001b5cdcb0a5e5cd382b21d9a07c279cdf90931f","family_id":"qf-rate-affine-average","seed":202610063067,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"4\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(4)x+(41). For distinct inputs 0 and 17, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"4/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=4. The exact average rate is 4/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0068","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4318b8da455fa0bc2a21fd698353f44e3a09b837d5e4fe4fd9e5000b9ce86725","family_id":"qf-rate-affine-average","seed":202610063068,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"5\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(5)x+(44). For distinct inputs 0 and -18, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"5/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=5. The exact average rate is 5/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0069","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b95380c6b411059d16caa21cc037cea9f10d5e949c0f5e843620ff8e4b99408","family_id":"qf-rate-affine-average","seed":202610063069,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"6\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(6)x+(47). For distinct inputs 0 and 19, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"6/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=6. The exact average rate is 6/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0070","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7569d65f6b76c12be228d30b375d962d8afea3fe916f406846266887c7f64d13","family_id":"qf-rate-affine-average","seed":202610063070,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"7\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(7)x+(50). For distinct inputs 0 and -20, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"7/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=7. The exact average rate is 7/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0071","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7f0306ee81e482d4aa28d5cdfd0acba4b18aa93d0c4c96c798ee4d117f3a048","family_id":"qf-rate-affine-average","seed":202610063071,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"8\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(8)x+(53). For distinct inputs 0 and 21, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"8/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=8. The exact average rate is 8/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0072","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a4cd68c3975b3a369df850765ab9ec312b449200e41e226e4c782d6f66a68b9","family_id":"qf-rate-affine-average","seed":202610063072,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"9\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(9)x+(56). For distinct inputs 0 and -22, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"9/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=9. The exact average rate is 9/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0073","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c21a108d4cc66eb12a2136ef67f6e2d6b0469d61f19978be39aca7999d064467","family_id":"qf-rate-affine-average","seed":202610063073,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"10\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(10)x+(59). For distinct inputs 0 and 23, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"10/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=10. The exact average rate is 10/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0074","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c63798157619e097f2323c35408362b0b9fff5f314bc8d477423b9bbec0575b7","family_id":"qf-rate-affine-average","seed":202610063074,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"11\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(11)x+(62). For distinct inputs 0 and -24, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"11/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=11. The exact average rate is 11/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0075","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05d13170fb7e11bef108b72bb4e4f4216441c86cbe251d2bb77d9781d9678a8e","family_id":"qf-rate-affine-average","seed":202610063075,"semantic_template":"qf-rate-affine-average/{\"inputs\":[\"12\"],\"operation\":\"affine_average\"}"},"fields":{"title":"Average Rate of a Real Affine Function","stem":"On the full real domain, f(x)=(12)x+(65). For distinct inputs 0 and 25, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"12/1","explanation":"The common constant cancels in the endpoint difference, leaving slope m=12. The exact average rate is 12/1, including zero slope. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Real Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0076","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"776b2f4b22b524fb1f0db8a2d11f633f74b543be351639719e7b22a17db20d1b","family_id":"qf-rate-quadratic-average","seed":202610063076,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"symmetric_endpoints\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(1)(x−(0))²+(0), with a≠0. For distinct inputs -1 and 1, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"0/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 0/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0077","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d915b93ab4104dbb651fbd0e0d9a7e999d85b4617f696671424e592c49e5614b","family_id":"qf-rate-quadratic-average","seed":202610063077,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-2\",\"1\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(-2)(x−(-11))²+(-11), with a≠0. For distinct inputs -9 and -12, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-2/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -2/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0078","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5799d1e1fae935335efc4ec6076c1fd78af9e176b77541d78ce5cb8b3deee90e","family_id":"qf-rate-quadratic-average","seed":202610063078,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"3\",\"2\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(3)(x−(-10))²+(-10), with a≠0. For distinct inputs -11 and -7, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"6/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 6/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0079","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a08ac85bda369e0a36734352f8240fbd302d4cd036a812cea5aab0c4be7e9f87","family_id":"qf-rate-quadratic-average","seed":202610063079,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-4\",\"3\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(-4)(x−(-9))²+(-9), with a≠0. For distinct inputs -5 and -10, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-12/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -12/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0080","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"337dceb54409f2a65ab964ca2fa9df6da7254fd769e567d4c7d9a8091af61a72","family_id":"qf-rate-quadratic-average","seed":202610063080,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"5\",\"4\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(5)(x−(-8))²+(-8), with a≠0. For distinct inputs -9 and -3, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"20/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 20/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0081","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11c9fd92c0cf52fb5c35ecb027cb4c2c81e9e99eee67c6d83ddc027abd2b952b","family_id":"qf-rate-quadratic-average","seed":202610063081,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-6\",\"5\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(-6)(x−(-7))²+(-7), with a≠0. For distinct inputs -1 and -8, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-30/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -30/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0082","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d40babe9a046a3f8a8127ecb97f2b547768e61eced0ccf2a6dc735aa22f2bc2","family_id":"qf-rate-quadratic-average","seed":202610063082,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"7\",\"6\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(7)(x−(-6))²+(-6), with a≠0. For distinct inputs -7 and 1, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"42/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 42/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0083","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f93c9e1984eb6026ac11685719c61ea43048f80faef57eb08ffc0b8cedc1320f","family_id":"qf-rate-quadratic-average","seed":202610063083,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-8\",\"7\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(-8)(x−(-5))²+(-5), with a≠0. For distinct inputs 3 and -6, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-56/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -56/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0084","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fa29244d42fc553735155c6151463a2df54c127d8e3e3a868a462cfd06d4261","family_id":"qf-rate-quadratic-average","seed":202610063084,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"9\",\"8\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(9)(x−(-4))²+(-4), with a≠0. For distinct inputs -5 and 5, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"72/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 72/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0085","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbfe0b13bdff055f27b986fdd1cbea31e397ebf15d70e9e58848672917a53863","family_id":"qf-rate-quadratic-average","seed":202610063085,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-10\",\"9\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(-10)(x−(-3))²+(-3), with a≠0. For distinct inputs 7 and -4, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-90/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -90/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0086","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97bcda5a51abde674336f77eab45c5b601863f65972aa061b189713ca4663810","family_id":"qf-rate-quadratic-average","seed":202610063086,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"11\",\"10\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(11)(x−(-2))²+(-2), with a≠0. For distinct inputs -3 and 9, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"110/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 110/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0087","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c04948b3b3d6ad1482dc81e51710b282c17cc2614c9a24c27aba7aea101f39f","family_id":"qf-rate-quadratic-average","seed":202610063087,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-12\",\"11\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(-12)(x−(-1))²+(-1), with a≠0. For distinct inputs 11 and -2, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-132/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -132/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0088","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"521f781ba2adbdd43bd0f9d5d5a7824246423f69eaf052dd110e64fd6c773691","family_id":"qf-rate-quadratic-average","seed":202610063088,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"13\",\"12\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(13)(x−(0))²+(0), with a≠0. For distinct inputs -1 and 13, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"156/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 156/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0089","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3645688a1595fe5848014efe6c744daa86a2be9db80b0afa9f1852d135262b41","family_id":"qf-rate-quadratic-average","seed":202610063089,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-14\",\"13\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(-14)(x−(1))²+(1), with a≠0. For distinct inputs 15 and 0, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-182/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -182/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0090","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a441d6866e364ee5889f7e0ca2aee383c020b0c314eeacbc5cd9982e3bdedf9","family_id":"qf-rate-quadratic-average","seed":202610063090,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"15\",\"14\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(15)(x−(2))²+(2), with a≠0. For distinct inputs 1 and 17, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"210/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 210/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0091","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce446525f37cf7cb6c3bc25a9c54ed58e3bb96e55fd462dab4383fb2bcd400f5","family_id":"qf-rate-quadratic-average","seed":202610063091,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-16\",\"15\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(-16)(x−(3))²+(3), with a≠0. For distinct inputs 19 and 2, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-240/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -240/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0092","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64f6b798a3201e4ebb6104982bac94a378abc197398611d11351a57ca2583db9","family_id":"qf-rate-quadratic-average","seed":202610063092,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"17\",\"16\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(17)(x−(4))²+(4), with a≠0. For distinct inputs 3 and 21, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"272/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 272/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0093","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a2081c1c218c32ea2cb1ee34d0c483687f7bfad276b5154c75f1ecbbd2437f9","family_id":"qf-rate-quadratic-average","seed":202610063093,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-18\",\"17\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(-18)(x−(5))²+(5), with a≠0. For distinct inputs 23 and 4, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-306/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -306/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0094","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93ece9efdd080cb8f7a9b427d698f66609838c602cce92ea4993a6a34514411a","family_id":"qf-rate-quadratic-average","seed":202610063094,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"19\",\"18\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(19)(x−(6))²+(6), with a≠0. For distinct inputs 5 and 25, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"342/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 342/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0095","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5b32f82776744c7294c2b32752afcae2f2e77cd3b66b6e278123f2e6c147057","family_id":"qf-rate-quadratic-average","seed":202610063095,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-20\",\"19\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(-20)(x−(7))²+(7), with a≠0. For distinct inputs 27 and 6, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-380/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -380/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0096","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3f8b99fa7d48d53daa9966c07e6fe16835565dba8690c658a98e8cff843c0ce","family_id":"qf-rate-quadratic-average","seed":202610063096,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"21\",\"20\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(21)(x−(8))²+(8), with a≠0. For distinct inputs 7 and 29, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"420/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 420/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0097","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99884570a8f0e9af953a98edb2571b21c49dcb3ce438e1edf2830433ae22e6bb","family_id":"qf-rate-quadratic-average","seed":202610063097,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-22\",\"21\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(-22)(x−(9))²+(9), with a≠0. For distinct inputs 31 and 8, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-462/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -462/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0098","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec4145177eb28794226239de39e89586218374d1c47850b0b77c342ba2204afc","family_id":"qf-rate-quadratic-average","seed":202610063098,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"23\",\"22\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(23)(x−(10))²+(10), with a≠0. For distinct inputs 9 and 33, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"506/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 506/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0099","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"631e0269533363500adc822a5165585efcb5c4f45ec332bf3a3f48e17f8c55d2","family_id":"qf-rate-quadratic-average","seed":202610063099,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-24\",\"23\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(-24)(x−(11))²+(11), with a≠0. For distinct inputs 35 and 10, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-552/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -552/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r03-rate-of-change-0100","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"509d8237a95bd7c603302305de42af8e6d3bd357921db42fad2f33abc10ba739","family_id":"qf-rate-quadratic-average","seed":202610063100,"semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"25\",\"24\"],\"operation\":\"quadratic_average\"}"},"fields":{"title":"Average Rate of a Vertex-Form Quadratic","stem":"On the full real domain, let q:R→R be q(x)=(25)(x−(12))²+(12), with a≠0. For distinct inputs 11 and 37, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"600/1","explanation":"The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 600/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative."},"skills":["Average Rate of a Vertex-Form Quadratic"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}