{ "schemaVersion": "1.0", "moduleId": "algebra-manipulation-0580-pilot", "pathways": { "simplify-expand": { "minimumDistinctItems": 19, "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "items": [ { "id": "simp-q01", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Simplify 2x + 3y + 5x − 9y.", "data": { "family": "collect", "misconceptionDistractor": "x", "misconceptionDistractors": [ { "value": "7x + 6y", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." }, { "value": "8x − 6y", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "7x − 6y", "difficulty": 1, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Collect the like terms: 7x − 6y.", "misconceptionCode": "am-unlike-merged", "options": [ "7x − 6y", "7x + 6y", "x", "8x − 6y" ], "correctIndex": 0 }, { "id": "simp-q02", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Simplify 4x − 2 + 3x + 5.", "data": { "family": "collect", "misconceptionDistractor": "7x − 3", "misconceptionDistractors": [ { "value": "10x", "code": "am-unlike-merged", "feedback": "Only LIKE terms merge: same letters, same powers. x and x² (or x and y) stay apart." }, { "value": "8x + 3", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "7x + 3", "difficulty": 1, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Collect the like terms: 7x + 3.", "misconceptionCode": "am-sign-lost", "options": [ "10x", "7x + 3", "7x − 3", "8x + 3" ], "correctIndex": 1 }, { "id": "simp-q03", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Simplify 6x + x − 4x.", "data": { "family": "collect", "misconceptionDistractor": "4x", "misconceptionDistractors": [ { "value": "x", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." }, { "value": "2x", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "3x", "difficulty": 1, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Collect the like terms: 3x.", "misconceptionCode": "am-arithmetic", "options": [ "x", "4x", "3x", "2x" ], "correctIndex": 2 }, { "id": "simp-q04", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Simplify 2x^2 + 3xy − 1 + 5x^2 − 9xy + 4.", "data": { "family": "collect", "misconceptionDistractor": "4x^2", "misconceptionDistractors": [ { "value": "−3x^2 − 12xy − 5", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." }, { "value": "8x^2 − 6xy + 3", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "7x^2 − 6xy + 3", "difficulty": 2, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Collect the like terms: 7x^2 − 6xy + 3.", "misconceptionCode": "am-unlike-merged", "options": [ "−3x^2 − 12xy − 5", "8x^2 − 6xy + 3", "4x^2", "7x^2 − 6xy + 3" ], "correctIndex": 3 }, { "id": "simp-q05", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Simplify 3x^2 − 2x + 2x^2 + 6x − x^2.", "data": { "family": "collect", "misconceptionDistractor": "4x^2 − 4x", "misconceptionDistractors": [ { "value": "8x^2", "code": "am-unlike-merged", "feedback": "Only LIKE terms merge: same letters, same powers. x and x² (or x and y) stay apart." }, { "value": "5x^2 + 4x", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "4x^2 + 4x", "difficulty": 2, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Collect the like terms: 4x^2 + 4x.", "misconceptionCode": "am-sign-lost", "options": [ "4x^2 + 4x", "8x^2", "5x^2 + 4x", "4x^2 − 4x" ], "correctIndex": 0 }, { "id": "simp-q06", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Simplify 5y + 2 − 3y + 1 + 2y.", "data": { "family": "collect", "misconceptionDistractor": "5y + 3", "misconceptionDistractors": [ { "value": "7y", "code": "am-unlike-merged", "feedback": "Only LIKE terms merge: same letters, same powers. x and x² (or x and y) stay apart." }, { "value": "4y − 3", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." } ] }, "correctAnswer": "4y + 3", "difficulty": 2, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Collect the like terms: 4y + 3.", "misconceptionCode": "am-arithmetic", "options": [ "7y", "4y + 3", "4y − 3", "5y + 3" ], "correctIndex": 1 }, { "id": "simp-q07", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Simplify 4x^2 + 3x − 2x^2 − 5x + x.", "data": { "family": "collect", "misconceptionDistractor": "x^2", "misconceptionDistractors": [ { "value": "2x^2 + 3x", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." }, { "value": "3x^2 − x", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "2x^2 − x", "difficulty": 3, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Collect the like terms: 2x^2 − x.", "misconceptionCode": "am-unlike-merged", "options": [ "x^2", "2x^2 + 3x", "2x^2 − x", "3x^2 − x" ], "correctIndex": 2 }, { "id": "simp-q08", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Simplify 3xy + 2y − xy + 4y + xy.", "data": { "family": "collect", "misconceptionDistractor": "3xy − 6y", "misconceptionDistractors": [ { "value": "9xy", "code": "am-unlike-merged", "feedback": "Only LIKE terms merge: same letters, same powers. x and x² (or x and y) stay apart." }, { "value": "4xy + 6y", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "3xy + 6y", "difficulty": 3, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Collect the like terms: 3xy + 6y.", "misconceptionCode": "am-sign-lost", "options": [ "3xy − 6y", "9xy", "4xy + 6y", "3xy + 6y" ], "correctIndex": 3 }, { "id": "simp-q09", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Expand 3x(2x − 4y).", "data": { "family": "expand-single", "misconceptionDistractor": "6x^2 − 4y", "misconceptionDistractors": [ { "value": "5x^2 − 4y", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." }, { "value": "6x^2 + 12xy", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." } ] }, "correctAnswer": "6x^2 − 12xy", "difficulty": 1, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Each term inside is multiplied by 3x: 6x^2 − 12xy.", "misconceptionCode": "am-partial-expansion", "options": [ "6x^2 − 12xy", "5x^2 − 4y", "6x^2 − 4y", "6x^2 + 12xy" ], "correctIndex": 0 }, { "id": "simp-q10", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Expand 2(3x + 5).", "data": { "family": "expand-single", "misconceptionDistractor": "5x + 5", "misconceptionDistractors": [ { "value": "6x + 5", "code": "am-partial-expansion", "feedback": "Every term in the first bracket multiplies EVERY term in the second — four products, not two." }, { "value": "6x − 5", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." } ] }, "correctAnswer": "6x + 10", "difficulty": 1, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Each term inside is multiplied by 2: 6x + 10.", "misconceptionCode": "am-arithmetic", "options": [ "6x + 5", "6x + 10", "5x + 5", "6x − 5" ], "correctIndex": 1 }, { "id": "simp-q11", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Expand 4x(x − 3).", "data": { "family": "expand-single", "misconceptionDistractor": "4x^2 + 12x", "misconceptionDistractors": [ { "value": "4x^2 − 3", "code": "am-partial-expansion", "feedback": "Every term in the first bracket multiplies EVERY term in the second — four products, not two." }, { "value": "5x^2 − 3", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "4x^2 − 12x", "difficulty": 2, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Each term inside is multiplied by 4x: 4x^2 − 12x.", "misconceptionCode": "am-sign-lost", "options": [ "4x^2 − 3", "5x^2 − 3", "4x^2 − 12x", "4x^2 + 12x" ], "correctIndex": 2 }, { "id": "simp-q12", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Expand 5y(x + 2y).", "data": { "family": "expand-single", "misconceptionDistractor": "10y^2 + x", "misconceptionDistractors": [ { "value": "7y^2 + x", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." }, { "value": "10y^2 − x", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." } ] }, "correctAnswer": "5xy + 10y^2", "difficulty": 2, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Each term inside is multiplied by 5y: 5xy + 10y^2.", "misconceptionCode": "am-partial-expansion", "options": [ "7y^2 + x", "10y^2 − x", "10y^2 + x", "5xy + 10y^2" ], "correctIndex": 3 }, { "id": "simp-q13", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Expand and simplify (2x + 1)(x − 4).", "data": { "family": "expand-double", "misconceptionDistractor": "2x^2 − 7x + 4", "misconceptionDistractors": [ { "value": "2x^2 − 8x", "code": "am-partial-expansion", "feedback": "Every term in the first bracket multiplies EVERY term in the second — four products, not two." }, { "value": "2x^2 − 6x − 4", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "2x^2 − 7x − 4", "difficulty": 2, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Four products, then collect: 2x^2 − 7x − 4.", "misconceptionCode": "am-sign-lost", "options": [ "2x^2 − 7x − 4", "2x^2 − 8x", "2x^2 − 6x − 4", "2x^2 − 7x + 4" ], "correctIndex": 0 }, { "id": "simp-q14", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Expand and simplify (x + 3)(x + 5).", "data": { "family": "expand-double", "misconceptionDistractor": "x^2 + 5x", "misconceptionDistractors": [ { "value": "x^2 + 9x + 15", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." }, { "value": "x^2 − 8x + 15", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." } ] }, "correctAnswer": "x^2 + 8x + 15", "difficulty": 1, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Four products, then collect: x^2 + 8x + 15.", "misconceptionCode": "am-partial-expansion", "options": [ "x^2 + 9x + 15", "x^2 + 8x + 15", "x^2 + 5x", "x^2 − 8x + 15" ], "correctIndex": 1 }, { "id": "simp-q15", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Expand and simplify (x + 2)(x + 2).", "data": { "family": "expand-double", "misconceptionDistractor": "x^2 + 4", "misconceptionDistractors": [ { "value": "x^2 + 2x", "code": "am-partial-expansion", "feedback": "Every term in the first bracket multiplies EVERY term in the second — four products, not two." }, { "value": "x^2 − 4x + 4", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." } ] }, "correctAnswer": "x^2 + 4x + 4", "difficulty": 2, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Four products, then collect: x^2 + 4x + 4.", "misconceptionCode": "am-square-sum", "options": [ "x^2 + 2x", "x^2 − 4x + 4", "x^2 + 4x + 4", "x^2 + 4" ], "correctIndex": 2 }, { "id": "simp-q16", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Expand and simplify (x − 3)(x − 3).", "data": { "family": "expand-double", "misconceptionDistractor": "x^2 + 9", "misconceptionDistractors": [ { "value": "x^2 − 3x", "code": "am-partial-expansion", "feedback": "Every term in the first bracket multiplies EVERY term in the second — four products, not two." }, { "value": "x^2 + 6x + 9", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." } ] }, "correctAnswer": "x^2 − 6x + 9", "difficulty": 3, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Four products, then collect: x^2 − 6x + 9.", "misconceptionCode": "am-square-sum", "options": [ "x^2 + 9", "x^2 − 3x", "x^2 + 6x + 9", "x^2 − 6x + 9" ], "correctIndex": 3 }, { "id": "simp-q17", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Expand and simplify (3x + y)(x − 4y).", "data": { "family": "expand-double", "misconceptionDistractor": "3x^2 + 11xy − 4y^2", "misconceptionDistractors": [ { "value": "3x^2 − 12xy", "code": "am-partial-expansion", "feedback": "Every term in the first bracket multiplies EVERY term in the second — four products, not two." }, { "value": "3x^2 − 11xy − 4y^2 + x", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "3x^2 − 11xy − 4y^2", "difficulty": 3, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Four products, then collect: 3x^2 − 11xy − 4y^2.", "misconceptionCode": "am-sign-lost", "options": [ "3x^2 − 11xy − 4y^2", "3x^2 + 11xy − 4y^2", "3x^2 − 12xy", "3x^2 − 11xy − 4y^2 + x" ], "correctIndex": 0 }, { "id": "simp-q18", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Expand and simplify (2x − 1)(3x + 2).", "data": { "family": "expand-double", "misconceptionDistractor": "6x^2 + 4x", "misconceptionDistractors": [ { "value": "6x^2 + x + 2", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." }, { "value": "6x^2 + 2x − 2", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "6x^2 + x − 2", "difficulty": 3, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Four products, then collect: 6x^2 + x − 2.", "misconceptionCode": "am-partial-expansion", "options": [ "6x^2 + 4x", "6x^2 + x − 2", "6x^2 + x + 2", "6x^2 + 2x − 2" ], "correctIndex": 1 }, { "id": "simp-q19", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Expand and simplify (x − 2)(x + 3)(2x + 1).", "data": { "family": "expand-triple", "misconceptionDistractor": "x^2 + x − 6", "misconceptionDistractors": [ { "value": "2x^3 + 3x^2 + 11x + 6", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." }, { "value": "2x^3 + 3x^2 − 11x − 4", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "2x^3 + 3x^2 − 11x − 6", "difficulty": 4, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Multiply two brackets first, then by the third: 2x^3 + 3x^2 − 11x − 6.", "misconceptionCode": "am-partial-expansion", "options": [ "2x^3 + 3x^2 + 11x + 6", "x^2 + x − 6", "2x^3 + 3x^2 − 11x − 6", "2x^3 + 3x^2 − 11x − 4" ], "correctIndex": 2 }, { "id": "simp-q20", "objectiveRecordIds": [ "0580-core-C2.2-1", "0580-core-C2.2-2", "0580-extended-E2.2-1", "0580-extended-E2.2-2" ], "prompt": "Expand and simplify (x + 1)(x + 1)(x − 1).", "data": { "family": "expand-triple", "misconceptionDistractor": "x^2 + 2x + 1", "misconceptionDistractors": [ { "value": "x^3 + x^2 + x + 1", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." }, { "value": "x^3 + x^2 − x + 1", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "x^3 + x^2 − x − 1", "difficulty": 4, "hint": "Collect LIKE terms only: same letters, same powers. To expand, every term in one bracket multiplies EVERY term in the other — the grid has a cell for each product, and the sign travels with its term. (a + b)² = a² + 2ab + b², never a² + b².", "explanation": "Multiply two brackets first, then by the third: x^3 + x^2 − x − 1.", "misconceptionCode": "am-square-sum", "options": [ "x^3 + x^2 + x + 1", "x^2 + 2x + 1", "x^3 + x^2 − x + 1", "x^3 + x^2 − x − 1" ], "correctIndex": 3 } ] }, "factorise": { "minimumDistinctItems": 20, "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "items": [ { "id": "fact-q01", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise fully 9x^2 + 15xy.", "data": { "family": "common", "misconceptionDistractor": "3(3x^2 + 5xy)", "misconceptionDistractors": [ { "value": "x(9x + 15y)", "code": "am-not-fully", "feedback": "Factorise FULLY: there is still a common factor left inside the bracket." }, { "value": "3x(4x + 5y)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "3x(3x + 5y)", "difficulty": 1, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "The greatest common factor is 3x: 3x(3x + 5y).", "misconceptionCode": "am-letter-dropped", "options": [ "3x(3x + 5y)", "x(9x + 15y)", "3(3x^2 + 5xy)", "3x(4x + 5y)" ], "correctIndex": 0 }, { "id": "fact-q02", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise fully 6x + 12.", "data": { "family": "common", "misconceptionDistractor": "3(2x + 4)", "misconceptionDistractors": [ { "value": "6(2x + 2)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." }, { "value": "2(3x + 6)", "code": "am-not-fully", "feedback": "Factorise FULLY: there is still a common factor left inside the bracket." } ] }, "correctAnswer": "6(x + 2)", "difficulty": 1, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "The greatest common factor is 6: 6(x + 2).", "misconceptionCode": "am-not-fully", "options": [ "6(2x + 2)", "6(x + 2)", "3(2x + 4)", "2(3x + 6)" ], "correctIndex": 1 }, { "id": "fact-q03", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise fully 8x^2 + 4x.", "data": { "family": "common", "misconceptionDistractor": "4(2x^2 + x)", "misconceptionDistractors": [ { "value": "2x(4x + 2)", "code": "am-not-fully", "feedback": "Factorise FULLY: there is still a common factor left inside the bracket." }, { "value": "4x(3x + 1)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "4x(2x + 1)", "difficulty": 1, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "The greatest common factor is 4x: 4x(2x + 1).", "misconceptionCode": "am-letter-dropped", "options": [ "2x(4x + 2)", "4(2x^2 + x)", "4x(2x + 1)", "4x(3x + 1)" ], "correctIndex": 2 }, { "id": "fact-q04", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise fully 10x^2y + 15xy^2.", "data": { "family": "common", "misconceptionDistractor": "xy(10x + 15y)", "misconceptionDistractors": [ { "value": "5(2x^2y + 3xy^2)", "code": "am-letter-dropped", "feedback": "The common factor includes the LETTERS too: take out the highest shared power of every shared letter." }, { "value": "5xy(3x + 3y)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "5xy(2x + 3y)", "difficulty": 2, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "The greatest common factor is 5xy: 5xy(2x + 3y).", "misconceptionCode": "am-not-fully", "options": [ "5(2x^2y + 3xy^2)", "5xy(3x + 3y)", "xy(10x + 15y)", "5xy(2x + 3y)" ], "correctIndex": 3 }, { "id": "fact-q05", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise fully 14x^3 + 21x^2.", "data": { "family": "common", "misconceptionDistractor": "7(2x^3 + 3x^2)", "misconceptionDistractors": [ { "value": "x^2(14x + 21)", "code": "am-not-fully", "feedback": "Factorise FULLY: there is still a common factor left inside the bracket." }, { "value": "7x^2(3x + 3)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "7x^2(2x + 3)", "difficulty": 2, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "The greatest common factor is 7x^2: 7x^2(2x + 3).", "misconceptionCode": "am-letter-dropped", "options": [ "7x^2(2x + 3)", "x^2(14x + 21)", "7x^2(3x + 3)", "7(2x^3 + 3x^2)" ], "correctIndex": 0 }, { "id": "fact-q06", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise fully 18x^2 + 12xy + 6x.", "data": { "family": "common", "misconceptionDistractor": "3x(6x + 4y + 2)", "misconceptionDistractors": [ { "value": "6(3x^2 + 2xy + x)", "code": "am-letter-dropped", "feedback": "The common factor includes the LETTERS too: take out the highest shared power of every shared letter." }, { "value": "6x(4x + 2y + 1)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "6x(3x + 2y + 1)", "difficulty": 3, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "The greatest common factor is 6x: 6x(3x + 2y + 1).", "misconceptionCode": "am-not-fully", "options": [ "3x(6x + 4y + 2)", "6x(3x + 2y + 1)", "6(3x^2 + 2xy + x)", "6x(4x + 2y + 1)" ], "correctIndex": 1 }, { "id": "fact-q07", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise fully 5x + 10y.", "data": { "family": "grouping", "misconceptionDistractor": "5(x + y)", "misconceptionDistractors": [ { "value": "6(x + y)", "code": "am-not-fully", "feedback": "Factorise FULLY: there is still a common factor left inside the bracket." }, { "value": "6(x + 2y)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "5(x + 2y)", "difficulty": 3, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "Group the x terms and the y terms: x(5) + 2y(5) = 5(x + 2y).", "misconceptionCode": "am-wrong-pattern", "options": [ "5(x + y)", "6(x + y)", "5(x + 2y)", "6(x + 2y)" ], "correctIndex": 2 }, { "id": "fact-q08", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise fully 5x + 15y.", "data": { "family": "grouping", "misconceptionDistractor": "6(x + 3y)", "misconceptionDistractors": [ { "value": "5(x + y)", "code": "am-wrong-pattern", "feedback": "Check the pattern: a² − b² = (a + b)(a − b) — two DIFFERENT brackets, not a squared bracket." }, { "value": "4(x + 3y)", "code": "am-not-fully", "feedback": "Factorise FULLY: there is still a common factor left inside the bracket." } ] }, "correctAnswer": "5(x + 3y)", "difficulty": 3, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "Group the x terms and the y terms: x(5) + 3y(5) = 5(x + 3y).", "misconceptionCode": "am-arithmetic", "options": [ "6(x + 3y)", "5(x + y)", "4(x + 3y)", "5(x + 3y)" ], "correctIndex": 3 }, { "id": "fact-q09", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise x^2 − 9.", "data": { "family": "diff-squares", "misconceptionDistractor": "(x − 3)^2", "misconceptionDistractors": [ { "value": "(x + 3)(x + 3)", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." }, { "value": "(x + 9)(x − 1)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "(x + 3)(x − 3)", "difficulty": 2, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "Difference of two squares: (x + 3)(x − 3).", "misconceptionCode": "am-wrong-pattern", "options": [ "(x + 3)(x − 3)", "(x + 3)(x + 3)", "(x + 9)(x − 1)", "(x − 3)^2" ], "correctIndex": 0 }, { "id": "fact-q10", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise 4x^2 − 9y^2.", "data": { "family": "diff-squares", "misconceptionDistractor": "(2x − 3y)^2", "misconceptionDistractors": [ { "value": "(2x + 3y)(2x + 3y)", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." }, { "value": "(4x + 3y)(4x − 3y)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "(2x + 3y)(2x − 3y)", "difficulty": 3, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "Difference of two squares: (2x + 3y)(2x − 3y).", "misconceptionCode": "am-wrong-pattern", "options": [ "(2x − 3y)^2", "(2x + 3y)(2x − 3y)", "(2x + 3y)(2x + 3y)", "(4x + 3y)(4x − 3y)" ], "correctIndex": 1 }, { "id": "fact-q11", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise x^2 − 25.", "data": { "family": "diff-squares", "misconceptionDistractor": "(x + 25)(x − 1)", "misconceptionDistractors": [ { "value": "(x − 5)^2", "code": "am-wrong-pattern", "feedback": "Check the pattern: a² − b² = (a + b)(a − b) — two DIFFERENT brackets, not a squared bracket." }, { "value": "(x + 5)(x + 5)", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." } ] }, "correctAnswer": "(x + 5)(x − 5)", "difficulty": 2, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "Difference of two squares: (x + 5)(x − 5).", "misconceptionCode": "am-arithmetic", "options": [ "(x − 5)^2", "(x + 5)(x + 5)", "(x + 5)(x − 5)", "(x + 25)(x − 1)" ], "correctIndex": 2 }, { "id": "fact-q12", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise 9x^2 − 4y^2.", "data": { "family": "diff-squares", "misconceptionDistractor": "(3x + 2y)(3x + 2y)", "misconceptionDistractors": [ { "value": "(3x − 2y)^2", "code": "am-wrong-pattern", "feedback": "Check the pattern: a² − b² = (a + b)(a − b) — two DIFFERENT brackets, not a squared bracket." }, { "value": "(9x + 2y)(9x − 2y)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "(3x + 2y)(3x − 2y)", "difficulty": 4, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "Difference of two squares: (3x + 2y)(3x − 2y).", "misconceptionCode": "am-sign-lost", "options": [ "(3x − 2y)^2", "(3x + 2y)(3x + 2y)", "(9x + 2y)(9x − 2y)", "(3x + 2y)(3x − 2y)" ], "correctIndex": 3 }, { "id": "fact-q13", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise x^2 + 6x + 9.", "data": { "family": "perfect-square", "misconceptionDistractor": "x^2 + 9", "misconceptionDistractors": [ { "value": "(x + 3)(x + 4)", "code": "am-pair-sum", "feedback": "The pair must MULTIPLY to the constant AND ADD to the x-coefficient — check both." }, { "value": "(x − 3)^2", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." } ] }, "correctAnswer": "(x + 3)^2", "difficulty": 2, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "Perfect square: 9 = 3² and 6 = 2 × 3: (x + 3)^2.", "misconceptionCode": "am-square-sum", "options": [ "(x + 3)^2", "(x + 3)(x + 4)", "(x − 3)^2", "x^2 + 9" ], "correctIndex": 0 }, { "id": "fact-q14", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise x^2 − 8x + 16.", "data": { "family": "perfect-square", "misconceptionDistractor": "(x + 4)^2", "misconceptionDistractors": [ { "value": "x^2 + 16", "code": "am-square-sum", "feedback": "(a + b)² is NOT a² + b²: the middle term 2ab comes from multiplying the brackets out." }, { "value": "(x − 4)(x − 3)", "code": "am-pair-sum", "feedback": "The pair must MULTIPLY to the constant AND ADD to the x-coefficient — check both." } ] }, "correctAnswer": "(x − 4)^2", "difficulty": 3, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "Perfect square: 16 = 4² and -8 = 2 × -4: (x − 4)^2.", "misconceptionCode": "am-sign-lost", "options": [ "(x + 4)^2", "(x − 4)^2", "x^2 + 16", "(x − 4)(x − 3)" ], "correctIndex": 1 }, { "id": "fact-q15", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise x^2 + 3x + 2.", "data": { "family": "trinomial", "misconceptionDistractor": "(x + 1)(x + 1)", "misconceptionDistractors": [ { "value": "(x − 1)(x − 2)", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." }, { "value": "(x + 1)(x + 3)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "(x + 1)(x + 2)", "difficulty": 2, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "1 and 2 multiply to 2 and add to 3: (x + 1)(x + 2).", "misconceptionCode": "am-pair-sum", "options": [ "(x − 1)(x − 2)", "(x + 1)(x + 3)", "(x + 1)(x + 2)", "(x + 1)(x + 1)" ], "correctIndex": 2 }, { "id": "fact-q16", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise x^2 + 5x + 6.", "data": { "family": "trinomial", "misconceptionDistractor": "(x + 1)(x + 6)", "misconceptionDistractors": [ { "value": "(x − 2)(x − 3)", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." }, { "value": "(x + 2)(x + 4)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "(x + 2)(x + 3)", "difficulty": 2, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "2 and 3 multiply to 6 and add to 5: (x + 2)(x + 3).", "misconceptionCode": "am-pair-sum", "options": [ "(x − 2)(x − 3)", "(x + 2)(x + 4)", "(x + 1)(x + 6)", "(x + 2)(x + 3)" ], "correctIndex": 3 }, { "id": "fact-q17", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise x^2 − 7x + 6.", "data": { "family": "trinomial", "misconceptionDistractor": "(x + 1)(x + 6)", "misconceptionDistractors": [ { "value": "(x + 2)(x + 3)", "code": "am-pair-sum", "feedback": "The pair must MULTIPLY to the constant AND ADD to the x-coefficient — check both." }, { "value": "(x − 1)(x − 5)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "(x − 1)(x − 6)", "difficulty": 3, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "-1 and -6 multiply to 6 and add to -7: (x − 1)(x − 6).", "misconceptionCode": "am-sign-lost", "options": [ "(x − 1)(x − 6)", "(x + 1)(x + 6)", "(x + 2)(x + 3)", "(x − 1)(x − 5)" ], "correctIndex": 0 }, { "id": "fact-q18", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise x^2 + x − 12.", "data": { "family": "trinomial", "misconceptionDistractor": "(x − 4)(x + 3)", "misconceptionDistractors": [ { "value": "(x + 1)(x − 12)", "code": "am-pair-sum", "feedback": "The pair must MULTIPLY to the constant AND ADD to the x-coefficient — check both." }, { "value": "(x + 4)(x − 2)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "(x + 4)(x − 3)", "difficulty": 3, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "4 and -3 multiply to -12 and add to 1: (x + 4)(x − 3).", "misconceptionCode": "am-sign-lost", "options": [ "(x − 4)(x + 3)", "(x + 4)(x − 3)", "(x + 1)(x − 12)", "(x + 4)(x − 2)" ], "correctIndex": 1 }, { "id": "fact-q19", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise x^2 + 7x + 10.", "data": { "family": "trinomial", "misconceptionDistractor": "(x + 1)(x + 10)", "misconceptionDistractors": [ { "value": "(x − 2)(x − 5)", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." }, { "value": "(x + 2)(x + 6)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "(x + 2)(x + 5)", "difficulty": 3, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "2 and 5 multiply to 10 and add to 7: (x + 2)(x + 5).", "misconceptionCode": "am-pair-sum", "options": [ "(x + 1)(x + 10)", "(x − 2)(x − 5)", "(x + 2)(x + 5)", "(x + 2)(x + 6)" ], "correctIndex": 2 }, { "id": "fact-q20", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise x^2 − 10x + 16.", "data": { "family": "trinomial", "misconceptionDistractor": "(x − 2)(x − 7)", "misconceptionDistractors": [ { "value": "(x + 2)(x + 8)", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." }, { "value": "(x + 1)(x + 16)", "code": "am-pair-sum", "feedback": "The pair must MULTIPLY to the constant AND ADD to the x-coefficient — check both." } ] }, "correctAnswer": "(x − 2)(x − 8)", "difficulty": 4, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "-2 and -8 multiply to 16 and add to -10: (x − 2)(x − 8).", "misconceptionCode": "am-arithmetic", "options": [ "(x + 2)(x + 8)", "(x − 2)(x − 7)", "(x + 1)(x + 16)", "(x − 2)(x − 8)" ], "correctIndex": 3 }, { "id": "fact-q21", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Factorise fully x^3 − 3x^2 + 2x.", "data": { "family": "cubic-common", "misconceptionDistractor": "x(x^2 − 3x + 2)", "misconceptionDistractors": [ { "value": "x(x + 2)(x + 1)", "code": "am-pair-sum", "feedback": "The pair must MULTIPLY to the constant AND ADD to the x-coefficient — check both." }, { "value": "x(x − 2)(x − 2)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "x(x − 2)(x − 1)", "difficulty": 4, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "Take out x first, then factorise the quadratic: x(x − 2)(x − 1).", "misconceptionCode": "am-not-fully", "options": [ "x(x − 2)(x − 1)", "x(x + 2)(x + 1)", "x(x^2 − 3x + 2)", "x(x − 2)(x − 2)" ], "correctIndex": 0 }, { "id": "fact-q22", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Write x^2 + 6x + 2 in the form (x + h)² + k.", "data": { "family": "complete-square", "misconceptionDistractor": "(x + 6)^2 − 34", "misconceptionDistractors": [ { "value": "(x + 3)^2 + 2", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." }, { "value": "(x − 3)^2 − 7", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." } ] }, "correctAnswer": "(x + 3)^2 − 7", "difficulty": 2, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "Half of 6 is 3; 3² = 9, so the constant adjusts by 2 − 9 = -7: (x + 3)^2 − 7.", "misconceptionCode": "am-half-not", "options": [ "(x + 3)^2 + 2", "(x + 3)^2 − 7", "(x − 3)^2 − 7", "(x + 6)^2 − 34" ], "correctIndex": 1 }, { "id": "fact-q23", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Write x^2 + 8x + 10 in the form (x + h)² + k.", "data": { "family": "complete-square", "misconceptionDistractor": "(x + 4)^2 + 10", "misconceptionDistractors": [ { "value": "(x + 8)^2 − 54", "code": "am-half-not", "feedback": "Completing the square: the bracket takes HALF the x-coefficient, then adjust the constant." }, { "value": "(x − 4)^2 − 6", "code": "am-sign-lost", "feedback": "Carry the sign WITH its term into every grid cell: a negative times a positive is negative." } ] }, "correctAnswer": "(x + 4)^2 − 6", "difficulty": 3, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "Half of 8 is 4; 4² = 16, so the constant adjusts by 10 − 16 = -6: (x + 4)^2 − 6.", "misconceptionCode": "am-arithmetic", "options": [ "(x + 4)^2 + 10", "(x + 8)^2 − 54", "(x + 4)^2 − 6", "(x − 4)^2 − 6" ], "correctIndex": 2 }, { "id": "fact-q24", "objectiveRecordIds": [ "0580-core-C2.2-3", "0580-extended-E2.2-3", "0580-extended-E2.2-4", "0580-extended-E2.2-5" ], "prompt": "Write x^2 − 4x + 5 in the form (x + h)² + k.", "data": { "family": "complete-square", "misconceptionDistractor": "(x + 2)^2 + 1", "misconceptionDistractors": [ { "value": "(x − 4)^2 − 11", "code": "am-half-not", "feedback": "Completing the square: the bracket takes HALF the x-coefficient, then adjust the constant." }, { "value": "(x − 2)^2 + 5", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ] }, "correctAnswer": "(x − 2)^2 + 1", "difficulty": 3, "hint": "Factorise FULLY: take out the greatest common factor — number AND letters. Then check the pattern: grouping pairs up; a² − b² = (a + b)(a − b); a perfect square has equal brackets; a trinomial's pair multiplies to the constant and adds to the x-coefficient. Completing the square: halve the x-coefficient, square it, adjust the constant.", "explanation": "Half of -4 is -2; -2² = 4, so the constant adjusts by 5 − 4 = 1: (x − 2)^2 + 1.", "misconceptionCode": "am-sign-lost", "options": [ "(x + 2)^2 + 1", "(x − 4)^2 − 11", "(x − 2)^2 + 5", "(x − 2)^2 + 1" ], "correctIndex": 3 } ] }, "change-subject": { "minimumDistinctItems": 20, "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "items": [ { "id": "subj-q01", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make x the subject of y = 3x + 5.", "data": { "family": "subject-once", "misconceptionDistractor": "x = (y + 5)/3", "misconceptionDistractors": [ { "value": "x = y/3 + 5", "code": "am-both-sides", "feedback": "Whatever you do, do it to BOTH sides, and to the WHOLE side — the other terms move with it, not just the first one." }, { "value": "x = (y − 6)/3", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ], "scenario": { "given": "y = 3x + 5", "subject": "x", "make": "x", "point": { "x": 2, "y": 11 }, "steps": [ "Start: y = 3x + 5.", "Undo +5: subtract 5 from both sides → y − 5 = 3x.", "Undo ×3: divide both sides by 3 → x = (y − 5)/3." ] } }, "correctAnswer": "x = (y − 5)/3", "difficulty": 1, "hint": "Undo one operation at a time with its inverse on BOTH sides: add and subtract undo each other, multiply and divide undo each other, and every other term stays on the other side.", "explanation": "Undo each operation with its inverse on both sides: x = (y − 5)/3. At y = 11, x = 2.", "misconceptionCode": "am-inverse-wrong", "options": [ "x = (y − 5)/3", "x = y/3 + 5", "x = (y + 5)/3", "x = (y − 6)/3" ], "correctIndex": 0 }, { "id": "subj-q02", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make x the subject of y = 2x + 7.", "data": { "family": "subject-once", "misconceptionDistractor": "x = (y + 7)/2", "misconceptionDistractors": [ { "value": "x = y/2 + 7", "code": "am-both-sides", "feedback": "Whatever you do, do it to BOTH sides, and to the WHOLE side — the other terms move with it, not just the first one." }, { "value": "x = (y − 7) × 2", "code": "am-inverse-wrong", "feedback": "Undo each operation with its INVERSE: adding is undone by subtracting, multiplying by dividing. The same inverse must be used on every step." } ], "scenario": { "given": "y = 2x + 7", "subject": "x", "make": "x", "point": { "x": 3, "y": 13 }, "steps": [ "Start: y = 2x + 7.", "Undo +7: subtract 7 from both sides → y − 7 = 2x.", "Undo ×2: divide both sides by 2 → x = (y − 7)/2." ] } }, "correctAnswer": "x = (y − 7)/2", "difficulty": 1, "hint": "Undo one operation at a time with its inverse on BOTH sides: add and subtract undo each other, multiply and divide undo each other, and every other term stays on the other side.", "explanation": "Undo each operation with its inverse on both sides: x = (y − 7)/2. At y = 13, x = 3.", "misconceptionCode": "am-inverse-wrong", "options": [ "x = y/2 + 7", "x = (y − 7)/2", "x = (y + 7)/2", "x = (y − 7) × 2" ], "correctIndex": 1 }, { "id": "subj-q03", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make x the subject of y = 5x − 4.", "data": { "family": "subject-once", "misconceptionDistractor": "x = (y − 4)/5", "misconceptionDistractors": [ { "value": "x = y/5 − 4", "code": "am-both-sides", "feedback": "Whatever you do, do it to BOTH sides, and to the WHOLE side — the other terms move with it, not just the first one." }, { "value": "x = (y + 4) × 5", "code": "am-inverse-wrong", "feedback": "Undo each operation with its INVERSE: adding is undone by subtracting, multiplying by dividing. The same inverse must be used on every step." } ], "scenario": { "given": "y = 5x − 4", "subject": "x", "make": "x", "point": { "x": 2, "y": 6 }, "steps": [ "Start: y = 5x − 4.", "Undo −4: add 4 to both sides → y + 4 = 5x.", "Undo ×5: divide both sides by 5 → x = (y + 4)/5." ] } }, "correctAnswer": "x = (y + 4)/5", "difficulty": 2, "hint": "Undo one operation at a time with its inverse on BOTH sides: add and subtract undo each other, multiply and divide undo each other, and every other term stays on the other side.", "explanation": "Undo each operation with its inverse on both sides: x = (y + 4)/5. At y = 6, x = 2.", "misconceptionCode": "am-sign-move", "options": [ "x = y/5 − 4", "x = (y + 4) × 5", "x = (y + 4)/5", "x = (y − 4)/5" ], "correctIndex": 2 }, { "id": "subj-q04", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make x the subject of y = 4x + 1.", "data": { "family": "subject-once", "misconceptionDistractor": "x = (y + 1)/4", "misconceptionDistractors": [ { "value": "x = y/4 + 1", "code": "am-both-sides", "feedback": "Whatever you do, do it to BOTH sides, and to the WHOLE side — the other terms move with it, not just the first one." }, { "value": "x = (y − 1)/2", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ], "scenario": { "given": "y = 4x + 1", "subject": "x", "make": "x", "point": { "x": 3, "y": 13 }, "steps": [ "Start: y = 4x + 1.", "Undo +1: subtract 1 from both sides → y − 1 = 4x.", "Undo ×4: divide both sides by 4 → x = (y − 1)/4." ] } }, "correctAnswer": "x = (y − 1)/4", "difficulty": 2, "hint": "Undo one operation at a time with its inverse on BOTH sides: add and subtract undo each other, multiply and divide undo each other, and every other term stays on the other side.", "explanation": "Undo each operation with its inverse on both sides: x = (y − 1)/4. At y = 13, x = 3.", "misconceptionCode": "am-inverse-wrong", "options": [ "x = y/4 + 1", "x = (y − 1)/2", "x = (y + 1)/4", "x = (y − 1)/4" ], "correctIndex": 3 }, { "id": "subj-q05", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make x the subject of y = x + 6.", "data": { "family": "subject-once", "misconceptionDistractor": "x = y + 6", "misconceptionDistractors": [ { "value": "x = y/6", "code": "am-both-sides", "feedback": "Whatever you do, do it to BOTH sides, and to the WHOLE side — the other terms move with it, not just the first one." }, { "value": "x = 6 − y", "code": "am-sign-move", "feedback": "A term that crosses the equals sign changes sign, and the constant must not change sides twice. Keep the sign with its term." } ], "scenario": { "given": "y = x + 6", "subject": "x", "make": "x", "point": { "x": 4, "y": 10 }, "steps": [ "Start: y = x + 6.", "Undo +6: subtract 6 from both sides → x = y − 6." ] } }, "correctAnswer": "x = y − 6", "difficulty": 1, "hint": "Undo one operation at a time with its inverse on BOTH sides: add and subtract undo each other, multiply and divide undo each other, and every other term stays on the other side.", "explanation": "Undo each operation with its inverse on both sides: x = y − 6. At y = 10, x = 4.", "misconceptionCode": "am-inverse-wrong", "options": [ "x = y − 6", "x = y/6", "x = y + 6", "x = 6 − y" ], "correctIndex": 0 }, { "id": "subj-q06", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make q the subject of p = 2q + 3.", "data": { "family": "subject-once", "misconceptionDistractor": "q = p/2 + 3", "misconceptionDistractors": [ { "value": "q = (p + 3)/2", "code": "am-inverse-wrong", "feedback": "Undo each operation with its INVERSE: adding is undone by subtracting, multiplying by dividing. The same inverse must be used on every step." }, { "value": "q = (p − 3) × 2", "code": "am-inverse-wrong", "feedback": "Undo each operation with its INVERSE: adding is undone by subtracting, multiplying by dividing. The same inverse must be used on every step." } ], "scenario": { "given": "p = 2q + 3", "subject": "q", "make": "q", "point": { "q": 5, "p": 13 }, "steps": [ "Start: p = 2q + 3.", "Undo +3: subtract 3 from both sides → p − 3 = 2q.", "Undo ×2: divide both sides by 2 → q = (p − 3)/2." ] } }, "correctAnswer": "q = (p − 3)/2", "difficulty": 2, "hint": "Undo one operation at a time with its inverse on BOTH sides: add and subtract undo each other, multiply and divide undo each other, and every other term stays on the other side.", "explanation": "Undo each operation with its inverse on both sides: q = (p − 3)/2. At p = 13, q = 5.", "misconceptionCode": "am-both-sides", "options": [ "q = (p + 3)/2", "q = (p − 3)/2", "q = (p − 3) × 2", "q = p/2 + 3" ], "correctIndex": 1 }, { "id": "subj-q07", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make t the subject of v = 4t − 3.", "data": { "family": "subject-once", "misconceptionDistractor": "t = (v − 3)/4", "misconceptionDistractors": [ { "value": "t = v/4 − 3", "code": "am-both-sides", "feedback": "Whatever you do, do it to BOTH sides, and to the WHOLE side — the other terms move with it, not just the first one." }, { "value": "t = (v + 3) × 4", "code": "am-inverse-wrong", "feedback": "Undo each operation with its INVERSE: adding is undone by subtracting, multiplying by dividing. The same inverse must be used on every step." } ], "scenario": { "given": "v = 4t − 3", "subject": "t", "make": "t", "point": { "t": 2, "v": 5 }, "steps": [ "Start: v = 4t − 3.", "Undo −3: add 3 to both sides → v + 3 = 4t.", "Undo ×4: divide both sides by 4 → t = (v + 3)/4." ] } }, "correctAnswer": "t = (v + 3)/4", "difficulty": 2, "hint": "Undo one operation at a time with its inverse on BOTH sides: add and subtract undo each other, multiply and divide undo each other, and every other term stays on the other side.", "explanation": "Undo each operation with its inverse on both sides: t = (v + 3)/4. At v = 5, t = 2.", "misconceptionCode": "am-sign-move", "options": [ "t = v/4 − 3", "t = (v + 3) × 4", "t = (v + 3)/4", "t = (v − 3)/4" ], "correctIndex": 2 }, { "id": "subj-q08", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make r the subject of s = 7r − 2.", "data": { "family": "subject-once", "misconceptionDistractor": "r = (s − 2)/7", "misconceptionDistractors": [ { "value": "r = s/7 − 2", "code": "am-both-sides", "feedback": "Whatever you do, do it to BOTH sides, and to the WHOLE side — the other terms move with it, not just the first one." }, { "value": "r = (s + 2) × 7", "code": "am-inverse-wrong", "feedback": "Undo each operation with its INVERSE: adding is undone by subtracting, multiplying by dividing. The same inverse must be used on every step." } ], "scenario": { "given": "s = 7r − 2", "subject": "r", "make": "r", "point": { "r": 3, "s": 19 }, "steps": [ "Start: s = 7r − 2.", "Undo −2: add 2 to both sides → s + 2 = 7r.", "Undo ×7: divide both sides by 7 → r = (s + 2)/7." ] } }, "correctAnswer": "r = (s + 2)/7", "difficulty": 3, "hint": "Undo one operation at a time with its inverse on BOTH sides: add and subtract undo each other, multiply and divide undo each other, and every other term stays on the other side.", "explanation": "Undo each operation with its inverse on both sides: r = (s + 2)/7. At s = 19, r = 3.", "misconceptionCode": "am-sign-move", "options": [ "r = (s − 2)/7", "r = s/7 − 2", "r = (s + 2) × 7", "r = (s + 2)/7" ], "correctIndex": 3 }, { "id": "subj-q09", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make n the subject of m = 8n + 1.", "data": { "family": "subject-once", "misconceptionDistractor": "n = (m + 1)/8", "misconceptionDistractors": [ { "value": "n = m/8 + 1", "code": "am-both-sides", "feedback": "Whatever you do, do it to BOTH sides, and to the WHOLE side — the other terms move with it, not just the first one." }, { "value": "n = (m − 1)/4", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ], "scenario": { "given": "m = 8n + 1", "subject": "n", "make": "n", "point": { "n": 2, "m": 17 }, "steps": [ "Start: m = 8n + 1.", "Undo +1: subtract 1 from both sides → m − 1 = 8n.", "Undo ×8: divide both sides by 8 → n = (m − 1)/8." ] } }, "correctAnswer": "n = (m − 1)/8", "difficulty": 3, "hint": "Undo one operation at a time with its inverse on BOTH sides: add and subtract undo each other, multiply and divide undo each other, and every other term stays on the other side.", "explanation": "Undo each operation with its inverse on both sides: n = (m − 1)/8. At m = 17, n = 2.", "misconceptionCode": "am-inverse-wrong", "options": [ "n = (m − 1)/8", "n = (m + 1)/8", "n = m/8 + 1", "n = (m − 1)/4" ], "correctIndex": 0 }, { "id": "subj-q10", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make g the subject of h = 5g + 2.", "data": { "family": "subject-once", "misconceptionDistractor": "g = h/5 + 2", "misconceptionDistractors": [ { "value": "g = (h + 2)/5", "code": "am-inverse-wrong", "feedback": "Undo each operation with its INVERSE: adding is undone by subtracting, multiplying by dividing. The same inverse must be used on every step." }, { "value": "g = (h − 2) × 5", "code": "am-inverse-wrong", "feedback": "Undo each operation with its INVERSE: adding is undone by subtracting, multiplying by dividing. The same inverse must be used on every step." } ], "scenario": { "given": "h = 5g + 2", "subject": "g", "make": "g", "point": { "g": 3, "h": 17 }, "steps": [ "Start: h = 5g + 2.", "Undo +2: subtract 2 from both sides → h − 2 = 5g.", "Undo ×5: divide both sides by 5 → g = (h − 2)/5." ] } }, "correctAnswer": "g = (h − 2)/5", "difficulty": 3, "hint": "Undo one operation at a time with its inverse on BOTH sides: add and subtract undo each other, multiply and divide undo each other, and every other term stays on the other side.", "explanation": "Undo each operation with its inverse on both sides: g = (h − 2)/5. At h = 17, g = 3.", "misconceptionCode": "am-both-sides", "options": [ "g = h/5 + 2", "g = (h − 2)/5", "g = (h + 2)/5", "g = (h − 2) × 5" ], "correctIndex": 1 }, { "id": "subj-q11", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make x the subject of 2x + y = 5x − 3.", "data": { "family": "subject-collect", "misconceptionDistractor": "x = (y + 3)/7", "misconceptionDistractors": [ { "value": "x = (y − 3)/3", "code": "am-sign-move", "feedback": "A term that crosses the equals sign changes sign, and the constant must not change sides twice. Keep the sign with its term." }, { "value": "x = y/3 + 3", "code": "am-both-sides", "feedback": "Whatever you do, do it to BOTH sides, and to the WHOLE side — the other terms move with it, not just the first one." } ], "scenario": { "given": "2x + y = 5x − 3", "subject": "x", "make": "x", "point": { "x": 2, "y": 3 }, "steps": [ "Start: 2x + y = 5x − 3.", "Collect the x terms: subtract 2x from both sides → y = 3x − 3.", "Undo −3: add 3 to both sides → y + 3 = 3x.", "Undo ×3: divide both sides by 3 → x = (y + 3)/3." ] } }, "correctAnswer": "x = (y + 3)/3", "difficulty": 2, "hint": "The subject appears more than once: collect its terms on one side FIRST, then undo the operations in reverse order.", "explanation": "Undo each operation with its inverse on both sides: x = (y + 3)/3. At y = 3, x = 2.", "misconceptionCode": "am-collect-first", "options": [ "x = (y − 3)/3", "x = y/3 + 3", "x = (y + 3)/3", "x = (y + 3)/7" ], "correctIndex": 2 }, { "id": "subj-q12", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make x the subject of 4x − 1 = x + y.", "data": { "family": "subject-collect", "misconceptionDistractor": "x = (y + 1)/5", "misconceptionDistractors": [ { "value": "x = (y − 1)/3", "code": "am-sign-move", "feedback": "A term that crosses the equals sign changes sign, and the constant must not change sides twice. Keep the sign with its term." }, { "value": "x = y/3 + 1", "code": "am-both-sides", "feedback": "Whatever you do, do it to BOTH sides, and to the WHOLE side — the other terms move with it, not just the first one." } ], "scenario": { "given": "4x − 1 = x + y", "subject": "x", "make": "x", "point": { "x": 2, "y": 5 }, "steps": [ "Start: 4x − 1 = x + y.", "Collect the x terms: subtract x from both sides → 3x − 1 = y.", "Undo −1: add 1 to both sides → 3x = y + 1.", "Undo ×3: divide both sides by 3 → x = (y + 1)/3." ] } }, "correctAnswer": "x = (y + 1)/3", "difficulty": 2, "hint": "The subject appears more than once: collect its terms on one side FIRST, then undo the operations in reverse order.", "explanation": "Undo each operation with its inverse on both sides: x = (y + 1)/3. At y = 5, x = 2.", "misconceptionCode": "am-collect-first", "options": [ "x = (y − 1)/3", "x = y/3 + 1", "x = (y + 1)/5", "x = (y + 1)/3" ], "correctIndex": 3 }, { "id": "subj-q13", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make x the subject of 3x + 2y = 5x.", "data": { "family": "subject-collect", "misconceptionDistractor": "x = 2y", "misconceptionDistractors": [ { "value": "x = y/2", "code": "am-both-sides", "feedback": "Whatever you do, do it to BOTH sides, and to the WHOLE side — the other terms move with it, not just the first one." }, { "value": "x = −y", "code": "am-sign-move", "feedback": "A term that crosses the equals sign changes sign, and the constant must not change sides twice. Keep the sign with its term." } ], "scenario": { "given": "3x + 2y = 5x", "subject": "x", "make": "x", "point": { "x": 3, "y": 3 }, "steps": [ "Start: 3x + 2y = 5x.", "Collect the x terms: subtract 3x from both sides → 2y = 2x.", "Divide both sides by 2 → y = x, so x = y." ] } }, "correctAnswer": "x = y", "difficulty": 3, "hint": "The subject appears more than once: collect its terms on one side FIRST, then undo the operations in reverse order.", "explanation": "Undo each operation with its inverse on both sides: x = y. At y = 3, x = 3.", "misconceptionCode": "am-collect-first", "options": [ "x = y", "x = 2y", "x = y/2", "x = −y" ], "correctIndex": 0 }, { "id": "subj-q14", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make x the subject of 5x − 2 = 2x + 2y.", "data": { "family": "subject-collect", "misconceptionDistractor": "x = (2y + 2)/7", "misconceptionDistractors": [ { "value": "x = (2y − 2)/3", "code": "am-sign-move", "feedback": "A term that crosses the equals sign changes sign, and the constant must not change sides twice. Keep the sign with its term." }, { "value": "x = 2y/3 + 2", "code": "am-both-sides", "feedback": "Whatever you do, do it to BOTH sides, and to the WHOLE side — the other terms move with it, not just the first one." } ], "scenario": { "given": "5x − 2 = 2x + 2y", "subject": "x", "make": "x", "point": { "x": 2, "y": 2 }, "steps": [ "Start: 5x − 2 = 2x + 2y.", "Collect the x terms: subtract 2x from both sides → 3x − 2 = 2y.", "Undo −2: add 2 to both sides → 3x = 2y + 2.", "Undo ×3: divide both sides by 3 → x = (2y + 2)/3." ] } }, "correctAnswer": "x = (2y + 2)/3", "difficulty": 3, "hint": "The subject appears more than once: collect its terms on one side FIRST, then undo the operations in reverse order.", "explanation": "Undo each operation with its inverse on both sides: x = (2y + 2)/3. At y = 2, x = 2.", "misconceptionCode": "am-collect-first", "options": [ "x = (2y + 2)/7", "x = (2y + 2)/3", "x = (2y − 2)/3", "x = 2y/3 + 2" ], "correctIndex": 1 }, { "id": "subj-q15", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make x the subject of y = 3x + 2x − 7.", "data": { "family": "subject-collect", "misconceptionDistractor": "x = (y + 7)/3", "misconceptionDistractors": [ { "value": "x = (y − 7)/5", "code": "am-sign-move", "feedback": "A term that crosses the equals sign changes sign, and the constant must not change sides twice. Keep the sign with its term." }, { "value": "x = (y + 7)/6", "code": "am-collect-first", "feedback": "The subject appears more than once: collect its terms on one side FIRST, then divide by the coefficient that is left." } ], "scenario": { "given": "y = 3x + 2x − 7", "subject": "x", "make": "x", "point": { "x": 2, "y": 3 }, "steps": [ "Start: y = 3x + 2x − 7.", "Collect the x terms on the right: 3x + 2x = 5x → y = 5x − 7.", "Undo −7: add 7 to both sides → y + 7 = 5x.", "Undo ×5: divide both sides by 5 → x = (y + 7)/5." ] } }, "correctAnswer": "x = (y + 7)/5", "difficulty": 3, "hint": "The subject appears more than once: collect its terms on one side FIRST, then undo the operations in reverse order.", "explanation": "Undo each operation with its inverse on both sides: x = (y + 7)/5. At y = 3, x = 2.", "misconceptionCode": "am-collect-first", "options": [ "x = (y + 7)/3", "x = (y − 7)/5", "x = (y + 7)/5", "x = (y + 7)/6" ], "correctIndex": 2 }, { "id": "subj-q16", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make w the subject of h = 4w + 3w + 5.", "data": { "family": "subject-collect", "misconceptionDistractor": "w = (h − 5)/4", "misconceptionDistractors": [ { "value": "w = (h − 5)/3", "code": "am-collect-first", "feedback": "The subject appears more than once: collect its terms on one side FIRST, then divide by the coefficient that is left." }, { "value": "w = (h + 5)/7", "code": "am-sign-move", "feedback": "A term that crosses the equals sign changes sign, and the constant must not change sides twice. Keep the sign with its term." } ], "scenario": { "given": "h = 4w + 3w + 5", "subject": "w", "make": "w", "point": { "w": 2, "h": 19 }, "steps": [ "Start: h = 4w + 3w + 5.", "Collect the w terms: 4w + 3w = 7w → h = 7w + 5.", "Undo +5: subtract 5 from both sides → h − 5 = 7w.", "Undo ×7: divide both sides by 7 → w = (h − 5)/7." ] } }, "correctAnswer": "w = (h − 5)/7", "difficulty": 3, "hint": "The subject appears more than once: collect its terms on one side FIRST, then undo the operations in reverse order.", "explanation": "Undo each operation with its inverse on both sides: w = (h − 5)/7. At h = 19, w = 2.", "misconceptionCode": "am-collect-first", "options": [ "w = (h − 5)/4", "w = (h − 5)/3", "w = (h + 5)/7", "w = (h − 5)/7" ], "correctIndex": 3 }, { "id": "subj-q17", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make q the subject of p = 6q − 2q + 4.", "data": { "family": "subject-collect", "misconceptionDistractor": "q = (p − 4)/6", "misconceptionDistractors": [ { "value": "q = (p − 4)/2", "code": "am-collect-first", "feedback": "The subject appears more than once: collect its terms on one side FIRST, then divide by the coefficient that is left." }, { "value": "q = (p + 4)/4", "code": "am-sign-move", "feedback": "A term that crosses the equals sign changes sign, and the constant must not change sides twice. Keep the sign with its term." } ], "scenario": { "given": "p = 6q − 2q + 4", "subject": "q", "make": "q", "point": { "q": 3, "p": 16 }, "steps": [ "Start: p = 6q − 2q + 4.", "Collect the q terms: 6q − 2q = 4q → p = 4q + 4.", "Undo +4: subtract 4 from both sides → p − 4 = 4q.", "Undo ×4: divide both sides by 4 → q = (p − 4)/4." ] } }, "correctAnswer": "q = (p − 4)/4", "difficulty": 3, "hint": "The subject appears more than once: collect its terms on one side FIRST, then undo the operations in reverse order.", "explanation": "Undo each operation with its inverse on both sides: q = (p − 4)/4. At p = 16, q = 3.", "misconceptionCode": "am-collect-first", "options": [ "q = (p − 4)/4", "q = (p − 4)/6", "q = (p − 4)/2", "q = (p + 4)/4" ], "correctIndex": 0 }, { "id": "subj-q18", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make r the subject of h = 2r².", "data": { "family": "subject-power", "misconceptionDistractor": "r = √(h)/2", "misconceptionDistractors": [ { "value": "r = (h/2)²", "code": "am-power-order", "feedback": "Undo the operations in reverse order: remove the constant and the factor first; undo a square with a square root (or a square root with a square) LAST." }, { "value": "r = √(2h)", "code": "am-inverse-wrong", "feedback": "Undo each operation with its INVERSE: adding is undone by subtracting, multiplying by dividing. The same inverse must be used on every step." } ], "scenario": { "given": "h = 2r²", "subject": "r", "make": "r", "point": { "r": 3, "h": 18 }, "steps": [ "Start: h = 2r².", "Undo ×2: divide both sides by 2 → h/2 = r².", "Undo the square: square root both sides → r = √(h/2)." ] } }, "correctAnswer": "r = √(h/2)", "difficulty": 3, "hint": "Undo the operations in reverse order: remove the constant and the factor first, and undo a square with a square root (or a square root with a square) LAST.", "explanation": "Undo each operation with its inverse on both sides: r = √(h/2). At h = 18, r = 3.", "misconceptionCode": "am-both-sides", "options": [ "r = √(h)/2", "r = √(h/2)", "r = (h/2)²", "r = √(2h)" ], "correctIndex": 1 }, { "id": "subj-q19", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make b the subject of A = 3b².", "data": { "family": "subject-power", "misconceptionDistractor": "b = √(A)/3", "misconceptionDistractors": [ { "value": "b = (A/3)²", "code": "am-power-order", "feedback": "Undo the operations in reverse order: remove the constant and the factor first; undo a square with a square root (or a square root with a square) LAST." }, { "value": "b = √(3A)", "code": "am-inverse-wrong", "feedback": "Undo each operation with its INVERSE: adding is undone by subtracting, multiplying by dividing. The same inverse must be used on every step." } ], "scenario": { "given": "A = 3b²", "subject": "b", "make": "b", "point": { "b": 4, "A": 48 }, "steps": [ "Start: A = 3b².", "Undo ×3: divide both sides by 3 → A/3 = b².", "Undo the square: square root both sides → b = √(A/3)." ] } }, "correctAnswer": "b = √(A/3)", "difficulty": 3, "hint": "Undo the operations in reverse order: remove the constant and the factor first, and undo a square with a square root (or a square root with a square) LAST.", "explanation": "Undo each operation with its inverse on both sides: b = √(A/3). At A = 48, b = 4.", "misconceptionCode": "am-both-sides", "options": [ "b = √(A)/3", "b = (A/3)²", "b = √(A/3)", "b = √(3A)" ], "correctIndex": 2 }, { "id": "subj-q20", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make x the subject of y = x² + 5.", "data": { "family": "subject-power", "misconceptionDistractor": "x = √(y) − 5", "misconceptionDistractors": [ { "value": "x = √(y + 5)", "code": "am-sign-move", "feedback": "A term that crosses the equals sign changes sign, and the constant must not change sides twice. Keep the sign with its term." }, { "value": "x = (y − 5)²", "code": "am-power-order", "feedback": "Undo the operations in reverse order: remove the constant and the factor first; undo a square with a square root (or a square root with a square) LAST." } ], "scenario": { "given": "y = x² + 5", "subject": "x", "make": "x", "point": { "x": 4, "y": 21 }, "steps": [ "Start: y = x² + 5.", "Undo +5: subtract 5 from both sides → y − 5 = x².", "Undo the square: square root both sides → x = √(y − 5)." ] } }, "correctAnswer": "x = √(y − 5)", "difficulty": 3, "hint": "Undo the operations in reverse order: remove the constant and the factor first, and undo a square with a square root (or a square root with a square) LAST.", "explanation": "Undo each operation with its inverse on both sides: x = √(y − 5). At y = 21, x = 4.", "misconceptionCode": "am-both-sides", "options": [ "x = √(y) − 5", "x = √(y + 5)", "x = (y − 5)²", "x = √(y − 5)" ], "correctIndex": 3 }, { "id": "subj-q21", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make x the subject of y = x² − 7.", "data": { "family": "subject-power", "misconceptionDistractor": "x = √(y) + 7", "misconceptionDistractors": [ { "value": "x = √(7 − y)", "code": "am-sign-move", "feedback": "A term that crosses the equals sign changes sign, and the constant must not change sides twice. Keep the sign with its term." }, { "value": "x = (y + 7)²", "code": "am-power-order", "feedback": "Undo the operations in reverse order: remove the constant and the factor first; undo a square with a square root (or a square root with a square) LAST." } ], "scenario": { "given": "y = x² − 7", "subject": "x", "make": "x", "point": { "x": 3, "y": 2 }, "steps": [ "Start: y = x² − 7.", "Undo −7: add 7 to both sides → y + 7 = x².", "Undo the square: square root both sides → x = √(y + 7)." ] } }, "correctAnswer": "x = √(y + 7)", "difficulty": 3, "hint": "Undo the operations in reverse order: remove the constant and the factor first, and undo a square with a square root (or a square root with a square) LAST.", "explanation": "Undo each operation with its inverse on both sides: x = √(y + 7). At y = 2, x = 3.", "misconceptionCode": "am-both-sides", "options": [ "x = √(y + 7)", "x = √(y) + 7", "x = √(7 − y)", "x = (y + 7)²" ], "correctIndex": 0 }, { "id": "subj-q22", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make t the subject of s = 4√t.", "data": { "family": "subject-power", "misconceptionDistractor": "t = √(s/4)", "misconceptionDistractors": [ { "value": "t = (4s)²", "code": "am-both-sides", "feedback": "Whatever you do, do it to BOTH sides, and to the WHOLE side — the other terms move with it, not just the first one." }, { "value": "t = s/4", "code": "am-inverse-wrong", "feedback": "Undo each operation with its INVERSE: adding is undone by subtracting, multiplying by dividing. The same inverse must be used on every step." } ], "scenario": { "given": "s = 4√t", "subject": "t", "make": "t", "point": { "t": 9, "s": 12 }, "steps": [ "Start: s = 4√t.", "Undo ×4: divide both sides by 4 → s/4 = √t.", "Undo the square root: square both sides → t = (s/4)²." ] } }, "correctAnswer": "t = (s/4)²", "difficulty": 4, "hint": "Undo the operations in reverse order: remove the constant and the factor first, and undo a square with a square root (or a square root with a square) LAST.", "explanation": "Undo each operation with its inverse on both sides: t = (s/4)². At s = 12, t = 9.", "misconceptionCode": "am-power-order", "options": [ "t = (4s)²", "t = (s/4)²", "t = √(s/4)", "t = s/4" ], "correctIndex": 1 }, { "id": "subj-q23", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make w the subject of v = 5√w.", "data": { "family": "subject-power", "misconceptionDistractor": "w = √(v/5)", "misconceptionDistractors": [ { "value": "w = (5v)²", "code": "am-both-sides", "feedback": "Whatever you do, do it to BOTH sides, and to the WHOLE side — the other terms move with it, not just the first one." }, { "value": "w = v/5", "code": "am-inverse-wrong", "feedback": "Undo each operation with its INVERSE: adding is undone by subtracting, multiplying by dividing. The same inverse must be used on every step." } ], "scenario": { "given": "v = 5√w", "subject": "w", "make": "w", "point": { "w": 16, "v": 20 }, "steps": [ "Start: v = 5√w.", "Undo ×5: divide both sides by 5 → v/5 = √w.", "Undo the square root: square both sides → w = (v/5)²." ] } }, "correctAnswer": "w = (v/5)²", "difficulty": 4, "hint": "Undo the operations in reverse order: remove the constant and the factor first, and undo a square with a square root (or a square root with a square) LAST.", "explanation": "Undo each operation with its inverse on both sides: w = (v/5)². At v = 20, w = 16.", "misconceptionCode": "am-power-order", "options": [ "w = (5v)²", "w = √(v/5)", "w = (v/5)²", "w = v/5" ], "correctIndex": 2 }, { "id": "subj-q24", "objectiveRecordIds": [ "0580-core-C2.5-4", "0580-extended-E2.5-7" ], "prompt": "Make r the subject of P = 2r² + 3.", "data": { "family": "subject-power", "misconceptionDistractor": "r = √(P/2) − 3", "misconceptionDistractors": [ { "value": "r = √(P − 3)/2", "code": "am-power-order", "feedback": "Undo the operations in reverse order: remove the constant and the factor first; undo a square with a square root (or a square root with a square) LAST." }, { "value": "r = (P − 3)/2", "code": "am-inverse-wrong", "feedback": "Undo each operation with its INVERSE: adding is undone by subtracting, multiplying by dividing. The same inverse must be used on every step." } ], "scenario": { "given": "P = 2r² + 3", "subject": "r", "make": "r", "point": { "r": 2, "P": 11 }, "steps": [ "Start: P = 2r² + 3.", "Undo +3: subtract 3 from both sides → P − 3 = 2r².", "Undo ×2: divide both sides by 2 → (P − 3)/2 = r².", "Undo the square: square root both sides → r = √((P − 3)/2)." ] } }, "correctAnswer": "r = √((P − 3)/2)", "difficulty": 4, "hint": "Undo the operations in reverse order: remove the constant and the factor first, and undo a square with a square root (or a square root with a square) LAST.", "explanation": "Undo each operation with its inverse on both sides: r = √((P − 3)/2). At P = 11, r = 2.", "misconceptionCode": "am-both-sides", "options": [ "r = √(P − 3)/2", "r = √(P/2) − 3", "r = (P − 3)/2", "r = √((P − 3)/2)" ], "correctIndex": 3 } ] }, "rational-expressions": { "minimumDistinctItems": 20, "objectiveRecordIds": [ "0580-extended-E2.3-2" ], "items": [ { "id": "rat-q01", "objectiveRecordIds": [ "0580-extended-E2.3-2" ], "prompt": "Simplify (12x²y)/(8xy²).", "data": { "family": "rational-mono-cancel", "misconceptionDistractor": "(3x²y)/(2xy²)", "misconceptionDistractors": [ { "value": "(3y)/(2x)", "code": "am-cancel-wrong-pair", "feedback": "Cancel a factor on the TOP with the SAME factor on the BOTTOM — x with x, y with y, and one bracket with an identical bracket, never x with y." }, { "value": "(4x)/(2y)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ], "scenario": { "given": "(12x²y)/(8xy²)", "num": "12x²y", "den": "8xy²", "numParts": [ "12", "x²", "y" ], "denParts": [ "8", "x", "y²" ], "cancelPairs": [], "steps": [ "Numbers: 12 and 8 share 4 → divide both by 4: 12 ÷ 4 = 3, 8 ÷ 4 = 2.", "Letters: x: 2 − 1 = 1, so x stays on top; y: 2 − 1 = 1, so y stays below.", "Cancel: (12x²y)/(8xy²) = (3x)/(2y)." ], "result": "(3x)/(2y)", "point": { "x": 4, "y": 3 } } }, "correctAnswer": "(3x)/(2y)", "difficulty": 1, "hint": "Cancel the shared NUMBER factor (divide both by it) and each shared LETTER by subtracting its powers — a letter disappears only when the powers match exactly.", "explanation": "Cancel what the whole top and the whole bottom share: (12x²y)/(8xy²) = (3x)/(2y). At x = 4, y = 3 both give 2 — the form changed, the value did not.", "misconceptionCode": "am-power-subtract", "options": [ "(3x)/(2y)", "(3y)/(2x)", "(3x²y)/(2xy²)", "(4x)/(2y)" ], "correctIndex": 0 }, { "id": "rat-q02", "objectiveRecordIds": [ "0580-extended-E2.3-2" ], "prompt": "Simplify (10x³y)/(15xy).", "data": { "family": "rational-mono-cancel", "misconceptionDistractor": "(2x³y)/(3xy)", "misconceptionDistractors": [ { "value": "(3x²)/(3)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." }, { "value": "2/3", "code": "am-letters-vanished", "feedback": "A letter cancels ONLY where its powers are equal on top and bottom: x² over x still leaves one x. A letter that is left over must stay in the answer." } ], "scenario": { "given": "(10x³y)/(15xy)", "num": "10x³y", "den": "15xy", "numParts": [ "10", "x³", "y" ], "denParts": [ "15", "x", "y" ], "cancelPairs": [], "steps": [ "Numbers: 10 and 15 share 5 → divide both by 5: 10 ÷ 5 = 2, 15 ÷ 5 = 3.", "Letters: x: 3 − 1 = 2, so x² stays on top; y cancels completely (y top and bottom).", "Cancel: (10x³y)/(15xy) = (2x²)/(3)." ], "result": "(2x²)/(3)", "point": { "x": 3, "y": 2 } } }, "correctAnswer": "(2x²)/(3)", "difficulty": 1, "hint": "Cancel the shared NUMBER factor (divide both by it) and each shared LETTER by subtracting its powers — a letter disappears only when the powers match exactly.", "explanation": "Cancel what the whole top and the whole bottom share: (10x³y)/(15xy) = (2x²)/(3). 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At x = 2, y = 3 both give 2 — the form changed, the value did not.", "misconceptionCode": "am-power-subtract", "options": [ "(3y)/(3)", "(2y)/(3)", "2/3", "(2xy²)/(3xy)" ], "correctIndex": 1 }, { "id": "rat-q15", "objectiveRecordIds": [ "0580-extended-E2.3-2" ], "prompt": "Simplify (14x²y)/(21xy²).", "data": { "family": "rational-mono-cancel", "misconceptionDistractor": "(3x)/(3y)", "misconceptionDistractors": [ { "value": "(2y)/(3x)", "code": "am-cancel-wrong-pair", "feedback": "Cancel a factor on the TOP with the SAME factor on the BOTTOM — x with x, y with y, and one bracket with an identical bracket, never x with y." }, { "value": "(2x²y)/(3xy²)", "code": "am-power-subtract", "feedback": "Cancel matching letters by SUBTRACTING their powers: x² over x leaves one x on top, and y over y² leaves one y below. A letter disappears only when both sides have the SAME power." } ], "scenario": { "given": "(14x²y)/(21xy²)", "num": "14x²y", "den": "21xy²", "numParts": [ "14", "x²", "y" ], "denParts": [ "21", "x", "y²" ], "cancelPairs": [], "steps": [ "Numbers: 14 and 21 share 7 → divide both by 7: 14 ÷ 7 = 2, 21 ÷ 7 = 3.", "Letters: x: 2 − 1 = 1, so x stays on top; y: 2 − 1 = 1, so y stays below.", "Cancel: (14x²y)/(21xy²) = (2x)/(3y)." ], "result": "(2x)/(3y)", "point": { "x": 3, "y": 1 } } }, "correctAnswer": "(2x)/(3y)", "difficulty": 3, "hint": "Cancel the shared NUMBER factor (divide both by it) and each shared LETTER by subtracting its powers — a letter disappears only when the powers match exactly.", "explanation": "Cancel what the whole top and the whole bottom share: (14x²y)/(21xy²) = (2x)/(3y). At x = 3, y = 1 both give 2 — the form changed, the value did not.", "misconceptionCode": "am-arithmetic", "options": [ "(3x)/(3y)", "(2y)/(3x)", "(2x)/(3y)", "(2x²y)/(3xy²)" ], "correctIndex": 2 }, { "id": "rat-q16", "objectiveRecordIds": [ "0580-extended-E2.3-2" ], "prompt": "Simplify (20x³y²)/(15xy³).", "data": { "family": "rational-mono-cancel", "misconceptionDistractor": "(4x³y²)/(3xy³)", "misconceptionDistractors": [ { "value": "(4y)/(3x²)", "code": "am-cancel-wrong-pair", "feedback": "Cancel a factor on the TOP with the SAME factor on the BOTTOM — x with x, y with y, and one bracket with an identical bracket, never x with y." }, { "value": "(5x²)/(3y)", "code": "am-arithmetic", "feedback": "The method is right; check the arithmetic — signs and integers, kept exact." } ], "scenario": { "given": "(20x³y²)/(15xy³)", "num": "20x³y²", "den": "15xy³", "numParts": [ "20", "x³", "y²" ], "denParts": [ "15", "x", "y³" ], "cancelPairs": [], "steps": [ "Numbers: 20 and 15 share 5 → divide both by 5: 20 ÷ 5 = 4, 15 ÷ 5 = 3.", "Letters: x: 3 − 1 = 2, so x² stays on top; y: 3 − 2 = 1, so y stays below.", "Cancel: (20x³y²)/(15xy³) = (4x²)/(3y)." ], "result": "(4x²)/(3y)", "point": { "x": 3, "y": 2 } } }, "correctAnswer": "(4x²)/(3y)", "difficulty": 4, "hint": "Cancel the shared NUMBER factor (divide both by it) and each shared LETTER by subtracting its powers — a letter disappears only when the powers match exactly.", "explanation": "Cancel what the whole top and the whole bottom share: (20x³y²)/(15xy³) = (4x²)/(3y). At x = 3, y = 2 both give 6 — the form changed, the value did not.", "misconceptionCode": "am-power-subtract", "options": [ "(4y)/(3x²)", "(4x³y²)/(3xy³)", "(5x²)/(3y)", "(4x²)/(3y)" ], "correctIndex": 3 }, { "id": "rat-q17", "objectiveRecordIds": [ "0580-extended-E2.3-2" ], "prompt": "Simplify (x² + 5x + 6)/(x² − x − 6).", "data": { "family": "rational-pair-cancel", "misconceptionDistractor": "(5x + 6)/(−x − 6)", "misconceptionDistractors": [ { "value": "((x + 3)(x + 2))/((x − 3)(x + 2))", "code": "am-unfactored", "feedback": "Factorise the numerator and the denominator FIRST, then cancel the bracket they share. Factorising without cancelling leaves the fraction unsimplified." }, { "value": "(x + 3)/(x + 2)", "code": "am-cancel-wrong-pair", "feedback": "Cancel a factor on the TOP with the SAME factor on the BOTTOM — x with x, y with y, and one bracket with an identical bracket, never x with y." } ], "scenario": { "given": "(x² + 5x + 6)/(x² − x − 6)", "num": "x² + 5x + 6", "den": "x² − x − 6", "numParts": [ "(x + 3)", "(x + 2)" ], "denParts": [ "(x − 3)", "(x + 2)" ], "cancelPairs": [ [ "(x + 2)", "(x + 2)" ] ], "steps": [ "Factorise the top: x² + 5x + 6 = (x + 3)(x + 2).", "Factorise the bottom: x² − x − 6 = (x − 3)(x + 2).", "Cancel the shared bracket (x + 2): (x² + 5x + 6)/(x² − x − 6) = (x + 3)/(x − 3)." ], "result": "(x + 3)/(x − 3)", "point": { "x": 9 } } }, "correctAnswer": "(x + 3)/(x − 3)", "difficulty": 2, "hint": "Factorise the top and the bottom fully, then cancel a bracket that appears in BOTH. Never cancel a single term of a sum.", "explanation": "Factorise both sides, then cancel the bracket they share: (x² + 5x + 6)/(x² − x − 6) = (x + 3)/(x − 3). At x = 9 both give 2 — the form changed, the value did not.", "misconceptionCode": "am-cancel-across-sum", "options": [ "(x + 3)/(x − 3)", "((x + 3)(x + 2))/((x − 3)(x + 2))", "(x + 3)/(x + 2)", "(5x + 6)/(−x − 6)" ], "correctIndex": 0 }, { "id": "rat-q18", "objectiveRecordIds": [ "0580-extended-E2.3-2" ], "prompt": "Simplify (x² + 5x + 4)/(x² − x − 2).", "data": { "family": "rational-pair-cancel", "misconceptionDistractor": "((x + 4)(x + 1))/((x − 2)(x + 1))", "misconceptionDistractors": [ { "value": "(5x + 4)/(−x − 2)", "code": "am-cancel-across-sum", "feedback": "Cancel only a factor of the WHOLE numerator and the WHOLE denominator. One term of a sum cannot be cancelled on its own." }, { "value": "(x + 4)/(x + 1)", "code": "am-cancel-wrong-pair", "feedback": "Cancel a factor on the TOP with the SAME factor on the BOTTOM — x with x, y with y, and one bracket with an identical bracket, never x with y." } ], "scenario": { "given": "(x² + 5x + 4)/(x² − x − 2)", "num": "x² + 5x + 4", "den": "x² − x − 2", "numParts": [ "(x + 4)", "(x + 1)" ], "denParts": [ "(x − 2)", "(x + 1)" ], "cancelPairs": [ [ "(x + 1)", "(x + 1)" ] ], "steps": [ "Factorise the top: x² + 5x + 4 = (x + 4)(x + 1).", "Factorise the bottom: x² − x − 2 = (x − 2)(x + 1).", "Cancel the shared bracket (x + 1): (x² + 5x + 4)/(x² − x − 2) = (x + 4)/(x − 2)." ], "result": "(x + 4)/(x − 2)", "point": { "x": 5 } } }, "correctAnswer": "(x + 4)/(x − 2)", "difficulty": 2, "hint": "Factorise the top and the bottom fully, then cancel a bracket that appears in BOTH. Never cancel a single term of a sum.", "explanation": "Factorise both sides, then cancel the bracket they share: (x² + 5x + 4)/(x² − x − 2) = (x + 4)/(x − 2). At x = 5 both give 3 — the form changed, the value did not.", "misconceptionCode": "am-unfactored", "options": [ "(5x + 4)/(−x − 2)", "(x + 4)/(x − 2)", "(x + 4)/(x + 1)", "((x + 4)(x + 1))/((x − 2)(x + 1))" ], "correctIndex": 1 }, { "id": "rat-q19", "objectiveRecordIds": [ "0580-extended-E2.3-2" ], "prompt": "Simplify (x² + 8x + 15)/(x² + 2x − 3).", "data": { "family": "rational-pair-cancel", "misconceptionDistractor": "(x + 5)/(x + 3)", "misconceptionDistractors": [ { "value": "((x + 5)(x + 3))/((x − 1)(x + 3))", "code": "am-unfactored", "feedback": "Factorise the numerator and the denominator FIRST, then cancel the bracket they share. Factorising without cancelling leaves the fraction unsimplified." }, { "value": "(8x + 15)/(2x − 3)", "code": "am-cancel-across-sum", "feedback": "Cancel only a factor of the WHOLE numerator and the WHOLE denominator. One term of a sum cannot be cancelled on its own." } ], "scenario": { "given": "(x² + 8x + 15)/(x² + 2x − 3)", "num": "x² + 8x + 15", "den": "x² + 2x − 3", "numParts": [ "(x + 5)", "(x + 3)" ], "denParts": [ "(x − 1)", "(x + 3)" ], "cancelPairs": [ [ "(x + 3)", "(x + 3)" ] ], "steps": [ "Factorise the top: x² + 8x + 15 = (x + 5)(x + 3).", "Factorise the bottom: x² + 2x − 3 = (x − 1)(x + 3).", "Cancel the shared bracket (x + 3): (x² + 8x + 15)/(x² + 2x − 3) = (x + 5)/(x − 1)." ], "result": "(x + 5)/(x − 1)", "point": { "x": 3 } } }, "correctAnswer": "(x + 5)/(x − 1)", "difficulty": 3, "hint": "Factorise the top and the bottom fully, then cancel a bracket that appears in BOTH. Never cancel a single term of a sum.", "explanation": "Factorise both sides, then cancel the bracket they share: (x² + 8x + 15)/(x² + 2x − 3) = (x + 5)/(x − 1). At x = 3 both give 4 — the form changed, the value did not.", "misconceptionCode": "am-cancel-wrong-pair", "options": [ "(x + 5)/(x + 3)", "((x + 5)(x + 3))/((x − 1)(x + 3))", "(x + 5)/(x − 1)", "(8x + 15)/(2x − 3)" ], "correctIndex": 2 }, { "id": "rat-q20", "objectiveRecordIds": [ "0580-extended-E2.3-2" ], "prompt": "Simplify (x² + 3x + 2)/(x² − 4x − 5).", "data": { "family": "rational-pair-cancel", "misconceptionDistractor": "(3x + 2)/(−4x − 5)", "misconceptionDistractors": [ { "value": "((x + 2)(x + 1))/((x − 5)(x + 1))", "code": "am-unfactored", "feedback": "Factorise the numerator and the denominator FIRST, then cancel the bracket they share. Factorising without cancelling leaves the fraction unsimplified." }, { "value": "(x + 2)/(x + 1)", "code": "am-cancel-wrong-pair", "feedback": "Cancel a factor on the TOP with the SAME factor on the BOTTOM — x with x, y with y, and one bracket with an identical bracket, never x with y." } ], "scenario": { "given": "(x² + 3x + 2)/(x² − 4x − 5)", "num": "x² + 3x + 2", "den": "x² − 4x − 5", "numParts": [ "(x + 2)", "(x + 1)" ], "denParts": [ "(x − 5)", "(x + 1)" ], "cancelPairs": [ [ "(x + 1)", "(x + 1)" ] ], "steps": [ "Factorise the top: x² + 3x + 2 = (x + 2)(x + 1).", "Factorise the bottom: x² − 4x − 5 = (x − 5)(x + 1).", "Cancel the shared bracket (x + 1): (x² + 3x + 2)/(x² − 4x − 5) = (x + 2)/(x − 5)." ], "result": "(x + 2)/(x − 5)", "point": { "x": 6 } } }, "correctAnswer": "(x + 2)/(x − 5)", "difficulty": 2, "hint": "Factorise the top and the bottom fully, then cancel a bracket that appears in BOTH. Never cancel a single term of a sum.", "explanation": "Factorise both sides, then cancel the bracket they share: (x² + 3x + 2)/(x² − 4x − 5) = (x + 2)/(x − 5). At x = 6 both give 8 — the form changed, the value did not.", "misconceptionCode": "am-cancel-across-sum", "options": [ "((x + 2)(x + 1))/((x − 5)(x + 1))", "(x + 2)/(x + 1)", "(3x + 2)/(−4x − 5)", "(x + 2)/(x − 5)" ], "correctIndex": 3 }, { "id": "rat-q21", "objectiveRecordIds": [ "0580-extended-E2.3-2" ], "prompt": "Simplify (x² + 8x + 12)/(x² − x − 6).", "data": { "family": "rational-pair-cancel", "misconceptionDistractor": "((x + 6)(x + 2))/((x − 3)(x + 2))", "misconceptionDistractors": [ { "value": "(8x + 12)/(−x − 6)", "code": "am-cancel-across-sum", "feedback": "Cancel only a factor of the WHOLE numerator and the WHOLE denominator. One term of a sum cannot be cancelled on its own." }, { "value": "(x + 6)/(x + 2)", "code": "am-cancel-wrong-pair", "feedback": "Cancel a factor on the TOP with the SAME factor on the BOTTOM — x with x, y with y, and one bracket with an identical bracket, never x with y." } ], "scenario": { "given": "(x² + 8x + 12)/(x² − x − 6)", "num": "x² + 8x + 12", "den": "x² − x − 6", "numParts": [ "(x + 6)", "(x + 2)" ], "denParts": [ "(x − 3)", "(x + 2)" ], "cancelPairs": [ [ "(x + 2)", "(x + 2)" ] ], "steps": [ "Factorise the top: x² + 8x + 12 = (x + 6)(x + 2).", "Factorise the bottom: x² − x − 6 = (x − 3)(x + 2).", "Cancel the shared bracket (x + 2): (x² + 8x + 12)/(x² − x − 6) = (x + 6)/(x − 3)." ], "result": "(x + 6)/(x − 3)", "point": { "x": 6 } } }, "correctAnswer": "(x + 6)/(x − 3)", "difficulty": 3, "hint": "Factorise the top and the bottom fully, then cancel a bracket that appears in BOTH. Never cancel a single term of a sum.", "explanation": "Factorise both sides, then cancel the bracket they share: (x² + 8x + 12)/(x² − x − 6) = (x + 6)/(x − 3). At x = 6 both give 4 — the form changed, the value did not.", "misconceptionCode": "am-unfactored", "options": [ "(x + 6)/(x − 3)", "((x + 6)(x + 2))/((x − 3)(x + 2))", "(8x + 12)/(−x − 6)", "(x + 6)/(x + 2)" ], "correctIndex": 0 }, { "id": "rat-q22", "objectiveRecordIds": [ "0580-extended-E2.3-2" ], "prompt": "Simplify (x² + 6x + 5)/(x² − 3x − 4).", "data": { "family": "rational-pair-cancel", "misconceptionDistractor": "(x + 5)/(x + 1)", "misconceptionDistractors": [ { "value": "((x + 5)(x + 1))/((x − 4)(x + 1))", "code": "am-unfactored", "feedback": "Factorise the numerator and the denominator FIRST, then cancel the bracket they share. Factorising without cancelling leaves the fraction unsimplified." }, { "value": "(6x + 5)/(−3x − 4)", "code": "am-cancel-across-sum", "feedback": "Cancel only a factor of the WHOLE numerator and the WHOLE denominator. One term of a sum cannot be cancelled on its own." } ], "scenario": { "given": "(x² + 6x + 5)/(x² − 3x − 4)", "num": "x² + 6x + 5", "den": "x² − 3x − 4", "numParts": [ "(x + 5)", "(x + 1)" ], "denParts": [ "(x − 4)", "(x + 1)" ], "cancelPairs": [ [ "(x + 1)", "(x + 1)" ] ], "steps": [ "Factorise the top: x² + 6x + 5 = (x + 5)(x + 1).", "Factorise the bottom: x² − 3x − 4 = (x − 4)(x + 1).", "Cancel the shared bracket (x + 1): (x² + 6x + 5)/(x² − 3x − 4) = (x + 5)/(x − 4)." ], "result": "(x + 5)/(x − 4)", "point": { "x": 7 } } }, "correctAnswer": "(x + 5)/(x − 4)", "difficulty": 3, "hint": "Factorise the top and the bottom fully, then cancel a bracket that appears in BOTH. Never cancel a single term of a sum.", "explanation": "Factorise both sides, then cancel the bracket they share: (x² + 6x + 5)/(x² − 3x − 4) = (x + 5)/(x − 4). At x = 7 both give 4 — the form changed, the value did not.", "misconceptionCode": "am-cancel-wrong-pair", "options": [ "(x + 5)/(x + 1)", "(x + 5)/(x − 4)", "((x + 5)(x + 1))/((x − 4)(x + 1))", "(6x + 5)/(−3x − 4)" ], "correctIndex": 1 }, { "id": "rat-q23", "objectiveRecordIds": [ "0580-extended-E2.3-2" ], "prompt": "Simplify (x² + 10x + 24)/(x² + 2x − 8).", "data": { "family": "rational-pair-cancel", "misconceptionDistractor": "(10x + 24)/(2x − 8)", "misconceptionDistractors": [ { "value": "((x + 6)(x + 4))/((x − 2)(x + 4))", "code": "am-unfactored", "feedback": "Factorise the numerator and the denominator FIRST, then cancel the bracket they share. Factorising without cancelling leaves the fraction unsimplified." }, { "value": "(x + 6)/(x + 4)", "code": "am-cancel-wrong-pair", "feedback": "Cancel a factor on the TOP with the SAME factor on the BOTTOM — x with x, y with y, and one bracket with an identical bracket, never x with y." } ], "scenario": { "given": "(x² + 10x + 24)/(x² + 2x − 8)", "num": "x² + 10x + 24", "den": "x² + 2x − 8", "numParts": [ "(x + 6)", "(x + 4)" ], "denParts": [ "(x − 2)", "(x + 4)" ], "cancelPairs": [ [ "(x + 4)", "(x + 4)" ] ], "steps": [ "Factorise the top: x² + 10x + 24 = (x + 6)(x + 4).", "Factorise the bottom: x² + 2x − 8 = (x − 2)(x + 4).", "Cancel the shared bracket (x + 4): (x² + 10x + 24)/(x² + 2x − 8) = (x + 6)/(x − 2)." ], "result": "(x + 6)/(x − 2)", "point": { "x": 10 } } }, "correctAnswer": "(x + 6)/(x − 2)", "difficulty": 4, "hint": "Factorise the top and the bottom fully, then cancel a bracket that appears in BOTH. Never cancel a single term of a sum.", "explanation": "Factorise both sides, then cancel the bracket they share: (x² + 10x + 24)/(x² + 2x − 8) = (x + 6)/(x − 2). At x = 10 both give 2 — the form changed, the value did not.", "misconceptionCode": "am-cancel-across-sum", "options": [ "((x + 6)(x + 4))/((x − 2)(x + 4))", "(10x + 24)/(2x − 8)", "(x + 6)/(x − 2)", "(x + 6)/(x + 4)" ], "correctIndex": 2 }, { "id": "rat-q24", "objectiveRecordIds": [ "0580-extended-E2.3-2" ], "prompt": "Simplify (x² + 7x + 10)/(x² + x − 2).", "data": { "family": "rational-pair-cancel", "misconceptionDistractor": "((x + 5)(x + 2))/((x − 1)(x + 2))", "misconceptionDistractors": [ { "value": "(7x + 10)/(x − 2)", "code": "am-cancel-across-sum", "feedback": "Cancel only a factor of the WHOLE numerator and the WHOLE denominator. One term of a sum cannot be cancelled on its own." }, { "value": "(x + 5)/(x + 2)", "code": "am-cancel-wrong-pair", "feedback": "Cancel a factor on the TOP with the SAME factor on the BOTTOM — x with x, y with y, and one bracket with an identical bracket, never x with y." } ], "scenario": { "given": "(x² + 7x + 10)/(x² + x − 2)", "num": "x² + 7x + 10", "den": "x² + x − 2", "numParts": [ "(x + 5)", "(x + 2)" ], "denParts": [ "(x − 1)", "(x + 2)" ], "cancelPairs": [ [ "(x + 2)", "(x + 2)" ] ], "steps": [ "Factorise the top: x² + 7x + 10 = (x + 5)(x + 2).", "Factorise the bottom: x² + x − 2 = (x − 1)(x + 2).", "Cancel the shared bracket (x + 2): (x² + 7x + 10)/(x² + x − 2) = (x + 5)/(x − 1)." ], "result": "(x + 5)/(x − 1)", "point": { "x": 4 } } }, "correctAnswer": "(x + 5)/(x − 1)", "difficulty": 4, "hint": "Factorise the top and the bottom fully, then cancel a bracket that appears in BOTH. Never cancel a single term of a sum.", "explanation": "Factorise both sides, then cancel the bracket they share: (x² + 7x + 10)/(x² + x − 2) = (x + 5)/(x − 1). At x = 4 both give 3 — the form changed, the value did not.", "misconceptionCode": "am-unfactored", "options": [ "(7x + 10)/(x − 2)", "((x + 5)(x + 2))/((x − 1)(x + 2))", "(x + 5)/(x + 2)", "(x + 5)/(x − 1)" ], "correctIndex": 3 } ] }, "linear-nonlinear": { "minimumDistinctItems": 20, "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "items": [ { "id": "sys-q01", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x + 1 and y = x² − 6x + 11 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(3, 2) and (6, 5)", "misconceptionDistractors": [ { "value": "(−2, −1) and (−5, −4)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." }, { "value": "(2, 3)", "code": "am-one-root", "feedback": "A quadratic from substitution usually gives TWO x-values, and each one needs its own y. Keep both pairs." } ], "scenario": { "kind": "system-substitute", "shape": "parabola", "given": "y = x + 1 and y = x² − 6x + 11", "linear": "y = x + 1", "nonlinear": "y = x² − 6x + 11", "substituted": "x² − 7x + 10 = 0", "factored": "(x − 2)(x − 5) = 0", "pairs": [ "(2, 3)", "(5, 6)" ], "pairValues": [ [ 2, 3 ], [ 5, 6 ] ] } }, "correctAnswer": "(2, 3) and (5, 6)", "difficulty": 1, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − 7x + 10 = 0 → (x − 2)(x − 5) = 0, so x = 2 or x = 5. Each x goes back into y = x + 1: (2, 3) and (5, 6).", "misconceptionCode": "am-swap-pair", "options": [ "(2, 3) and (5, 6)", "(−2, −1) and (−5, −4)", "(3, 2) and (6, 5)", "(2, 3)" ], "correctIndex": 0 }, { "id": "sys-q02", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x + 2 and y = x² − 4x + 6 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(1, 3)", "misconceptionDistractors": [ { "value": "(3, 1) and (6, 4)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(−1, 1) and (−4, −2)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." } ], "scenario": { "kind": "system-substitute", "shape": "parabola", "given": "y = x + 2 and y = x² − 4x + 6", "linear": "y = x + 2", "nonlinear": "y = x² − 4x + 6", "substituted": "x² − 5x + 4 = 0", "factored": "(x − 1)(x − 4) = 0", "pairs": [ "(1, 3)", "(4, 6)" ], "pairValues": [ [ 1, 3 ], [ 4, 6 ] ] } }, "correctAnswer": "(1, 3) and (4, 6)", "difficulty": 1, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − 5x + 4 = 0 → (x − 1)(x − 4) = 0, so x = 1 or x = 4. Each x goes back into y = x + 2: (1, 3) and (4, 6).", "misconceptionCode": "am-one-root", "options": [ "(3, 1) and (6, 4)", "(1, 3) and (4, 6)", "(1, 3)", "(−1, 1) and (−4, −2)" ], "correctIndex": 1 }, { "id": "sys-q03", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = 2x + 1 and y = x² − 2x + 4 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(−1, −1) and (−3, −5)", "misconceptionDistractors": [ { "value": "(3, 1) and (7, 3)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(1, 3)", "code": "am-one-root", "feedback": "A quadratic from substitution usually gives TWO x-values, and each one needs its own y. Keep both pairs." } ], "scenario": { "kind": "system-substitute", "shape": "parabola", "given": "y = 2x + 1 and y = x² − 2x + 4", "linear": "y = 2x + 1", "nonlinear": "y = x² − 2x + 4", "substituted": "x² − 4x + 3 = 0", "factored": "(x − 1)(x − 3) = 0", "pairs": [ "(1, 3)", "(3, 7)" ], "pairValues": [ [ 1, 3 ], [ 3, 7 ] ] } }, "correctAnswer": "(1, 3) and (3, 7)", "difficulty": 2, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − 4x + 3 = 0 → (x − 1)(x − 3) = 0, so x = 1 or x = 3. Each x goes back into y = 2x + 1: (1, 3) and (3, 7).", "misconceptionCode": "am-root-sign", "options": [ "(3, 1) and (7, 3)", "(1, 3)", "(1, 3) and (3, 7)", "(−1, −1) and (−3, −5)" ], "correctIndex": 2 }, { "id": "sys-q04", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x − 1 and y = x² − 7 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(−2, 3) and (3, −2)", "misconceptionDistractors": [ { "value": "(−3, −2) and (2, 3)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(2, 1) and (−3, −4)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." } ], "scenario": { "kind": "system-substitute", "shape": "parabola", "given": "y = x − 1 and y = x² − 7", "linear": "y = x − 1", "nonlinear": "y = x² − 7", "substituted": "x² − x − 6 = 0", "factored": "(x + 2)(x − 3) = 0", "pairs": [ "(−2, −3)", "(3, 2)" ], "pairValues": [ [ -2, -3 ], [ 3, 2 ] ] } }, "correctAnswer": "(−2, −3) and (3, 2)", "difficulty": 2, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − x − 6 = 0 → (x + 2)(x − 3) = 0, so x = −2 or x = 3. Each x goes back into y = x − 1: (−2, −3) and (3, 2).", "misconceptionCode": "am-sign-pair", "options": [ "(−3, −2) and (2, 3)", "(2, 1) and (−3, −4)", "(−2, 3) and (3, −2)", "(−2, −3) and (3, 2)" ], "correctIndex": 3 }, { "id": "sys-q05", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = 3x + 2 and y = x² + 4 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(5, 1) and (8, 2)", "misconceptionDistractors": [ { "value": "(−1, −1) and (−2, −4)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." }, { "value": "(1, 5)", "code": "am-one-root", "feedback": "A quadratic from substitution usually gives TWO x-values, and each one needs its own y. Keep both pairs." } ], "scenario": { "kind": "system-substitute", "shape": "parabola", "given": "y = 3x + 2 and y = x² + 4", "linear": "y = 3x + 2", "nonlinear": "y = x² + 4", "substituted": "x² − 3x + 2 = 0", "factored": "(x − 1)(x − 2) = 0", "pairs": [ "(1, 5)", "(2, 8)" ], "pairValues": [ [ 1, 5 ], [ 2, 8 ] ] } }, "correctAnswer": "(1, 5) and (2, 8)", "difficulty": 3, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − 3x + 2 = 0 → (x − 1)(x − 2) = 0, so x = 1 or x = 2. Each x goes back into y = 3x + 2: (1, 5) and (2, 8).", "misconceptionCode": "am-swap-pair", "options": [ "(1, 5) and (2, 8)", "(5, 1) and (8, 2)", "(−1, −1) and (−2, −4)", "(1, 5)" ], "correctIndex": 0 }, { "id": "sys-q06", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x + 3 and y = x² + 5x + 6 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(3, 6) and (1, 4)", "misconceptionDistractors": [ { "value": "(0, −3) and (2, −1)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(−3, 0)", "code": "am-one-root", "feedback": "A quadratic from substitution usually gives TWO x-values, and each one needs its own y. Keep both pairs." } ], "scenario": { "kind": "system-substitute", "shape": "parabola", "given": "y = x + 3 and y = x² + 5x + 6", "linear": "y = x + 3", "nonlinear": "y = x² + 5x + 6", "substituted": "x² + 4x + 3 = 0", "factored": "(x + 3)(x + 1) = 0", "pairs": [ "(−3, 0)", "(−1, 2)" ], "pairValues": [ [ -3, 0 ], [ -1, 2 ] ] } }, "correctAnswer": "(−3, 0) and (−1, 2)", "difficulty": 3, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² + 4x + 3 = 0 → (x + 3)(x + 1) = 0, so x = −3 or x = −1. Each x goes back into y = x + 3: (−3, 0) and (−1, 2).", "misconceptionCode": "am-root-sign", "options": [ "(3, 6) and (1, 4)", "(−3, 0) and (−1, 2)", "(0, −3) and (2, −1)", "(−3, 0)" ], "correctIndex": 1 }, { "id": "sys-q07", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = 2x − 3 and y = x² − 4x + 5 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(2, 1) and (4, 6)", "misconceptionDistractors": [ { "value": "(1, 2) and (5, 4)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(−2, −7) and (−4, −11)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." } ], "scenario": { "kind": "system-substitute", "shape": "parabola", "given": "y = 2x − 3 and y = x² − 4x + 5", "linear": "y = 2x − 3", "nonlinear": "y = x² − 4x + 5", "substituted": "x² − 6x + 8 = 0", "factored": "(x − 2)(x − 4) = 0", "pairs": [ "(2, 1)", "(4, 5)" ], "pairValues": [ [ 2, 1 ], [ 4, 5 ] ] } }, "correctAnswer": "(2, 1) and (4, 5)", "difficulty": 3, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − 6x + 8 = 0 → (x − 2)(x − 4) = 0, so x = 2 or x = 4. Each x goes back into y = 2x − 3: (2, 1) and (4, 5).", "misconceptionCode": "am-arithmetic", "options": [ "(2, 1) and (4, 6)", "(1, 2) and (5, 4)", "(2, 1) and (4, 5)", "(−2, −7) and (−4, −11)" ], "correctIndex": 2 }, { "id": "sys-q08", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x + 4 and y = x² − x + 1 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(−1, 3)", "misconceptionDistractors": [ { "value": "(3, −1) and (7, 3)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(1, 5) and (−3, 1)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." } ], "scenario": { "kind": "system-substitute", "shape": "parabola", "given": "y = x + 4 and y = x² − x + 1", "linear": "y = x + 4", "nonlinear": "y = x² − x + 1", "substituted": "x² − 2x − 3 = 0", "factored": "(x + 1)(x − 3) = 0", "pairs": [ "(−1, 3)", "(3, 7)" ], "pairValues": [ [ -1, 3 ], [ 3, 7 ] ] } }, "correctAnswer": "(−1, 3) and (3, 7)", "difficulty": 1, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − 2x − 3 = 0 → (x + 1)(x − 3) = 0, so x = −1 or x = 3. Each x goes back into y = x + 4: (−1, 3) and (3, 7).", "misconceptionCode": "am-one-root", "options": [ "(3, −1) and (7, 3)", "(−1, 3)", "(1, 5) and (−3, 1)", "(−1, 3) and (3, 7)" ], "correctIndex": 3 }, { "id": "sys-q09", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = 4x + 1 and y = x² + x + 3 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(5, 1) and (9, 2)", "misconceptionDistractors": [ { "value": "(−1, −3) and (−2, −7)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." }, { "value": "(1, 5)", "code": "am-one-root", "feedback": "A quadratic from substitution usually gives TWO x-values, and each one needs its own y. Keep both pairs." } ], "scenario": { "kind": "system-substitute", "shape": "parabola", "given": "y = 4x + 1 and y = x² + x + 3", "linear": "y = 4x + 1", "nonlinear": "y = x² + x + 3", "substituted": "x² − 3x + 2 = 0", "factored": "(x − 1)(x − 2) = 0", "pairs": [ "(1, 5)", "(2, 9)" ], "pairValues": [ [ 1, 5 ], [ 2, 9 ] ] } }, "correctAnswer": "(1, 5) and (2, 9)", "difficulty": 4, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − 3x + 2 = 0 → (x − 1)(x − 2) = 0, so x = 1 or x = 2. Each x goes back into y = 4x + 1: (1, 5) and (2, 9).", "misconceptionCode": "am-swap-pair", "options": [ "(1, 5) and (2, 9)", "(−1, −3) and (−2, −7)", "(5, 1) and (9, 2)", "(1, 5)" ], "correctIndex": 0 }, { "id": "sys-q10", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x − 2 and y = x² − 8x + 16 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(3, 1) and (6, 5)", "misconceptionDistractors": [ { "value": "(1, 3) and (4, 6)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(−3, −5) and (−6, −8)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." } ], "scenario": { "kind": "system-substitute", "shape": "parabola", "given": "y = x − 2 and y = x² − 8x + 16", "linear": "y = x − 2", "nonlinear": "y = x² − 8x + 16", "substituted": "x² − 9x + 18 = 0", "factored": "(x − 3)(x − 6) = 0", "pairs": [ "(3, 1)", "(6, 4)" ], "pairValues": [ [ 3, 1 ], [ 6, 4 ] ] } }, "correctAnswer": "(3, 1) and (6, 4)", "difficulty": 4, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − 9x + 18 = 0 → (x − 3)(x − 6) = 0, so x = 3 or x = 6. Each x goes back into y = x − 2: (3, 1) and (6, 4).", "misconceptionCode": "am-arithmetic", "options": [ "(1, 3) and (4, 6)", "(3, 1) and (6, 4)", "(3, 1) and (6, 5)", "(−3, −5) and (−6, −8)" ], "correctIndex": 1 }, { "id": "sys-q11", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = 2x and y = x² + x − 2 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(1, 2) and (−2, −4)", "misconceptionDistractors": [ { "value": "(−2, −1) and (4, 2)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(−1, −2)", "code": "am-one-root", "feedback": "A quadratic from substitution usually gives TWO x-values, and each one needs its own y. Keep both pairs." } ], "scenario": { "kind": "system-substitute", "shape": "parabola", "given": "y = 2x and y = x² + x − 2", "linear": "y = 2x", "nonlinear": "y = x² + x − 2", "substituted": "x² − x − 2 = 0", "factored": "(x + 1)(x − 2) = 0", "pairs": [ "(−1, −2)", "(2, 4)" ], "pairValues": [ [ -1, -2 ], [ 2, 4 ] ] } }, "correctAnswer": "(−1, −2) and (2, 4)", "difficulty": 2, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − x − 2 = 0 → (x + 1)(x − 2) = 0, so x = −1 or x = 2. Each x goes back into y = 2x: (−1, −2) and (2, 4).", "misconceptionCode": "am-root-sign", "options": [ "(−2, −1) and (4, 2)", "(−1, −2)", "(−1, −2) and (2, 4)", "(1, 2) and (−2, −4)" ], "correctIndex": 2 }, { "id": "sys-q12", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x + 5 and y = x² + 2x + 3 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(−2, −3) and (1, −6)", "misconceptionDistractors": [ { "value": "(3, −2) and (6, 1)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(2, 7) and (−1, 4)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." } ], "scenario": { "kind": "system-substitute", "shape": "parabola", "given": "y = x + 5 and y = x² + 2x + 3", "linear": "y = x + 5", "nonlinear": "y = x² + 2x + 3", "substituted": "x² + x − 2 = 0", "factored": "(x + 2)(x − 1) = 0", "pairs": [ "(−2, 3)", "(1, 6)" ], "pairValues": [ [ -2, 3 ], [ 1, 6 ] ] } }, "correctAnswer": "(−2, 3) and (1, 6)", "difficulty": 4, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² + x − 2 = 0 → (x + 2)(x − 1) = 0, so x = −2 or x = 1. Each x goes back into y = x + 5: (−2, 3) and (1, 6).", "misconceptionCode": "am-sign-pair", "options": [ "(3, −2) and (6, 1)", "(−2, −3) and (1, −6)", "(2, 7) and (−1, 4)", "(−2, 3) and (1, 6)" ], "correctIndex": 3 }, { "id": "sys-q13", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x + 1 and x² + y² = 25 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(−3, −4) and (4, 3)", "misconceptionDistractors": [ { "value": "(4, 5) and (−3, −2)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." }, { "value": "(−4, −3)", "code": "am-one-root", "feedback": "A quadratic from substitution usually gives TWO x-values, and each one needs its own y. Keep both pairs." } ], "scenario": { "kind": "system-substitute", "shape": "circle", "given": "y = x + 1 and x² + y² = 25", "linear": "y = x + 1", "nonlinear": "x² + y² = 25", "substituted": "x² + x − 12 = 0", "factored": "(x + 4)(x − 3) = 0", "pairs": [ "(−4, −3)", "(3, 4)" ], "pairValues": [ [ -4, -3 ], [ 3, 4 ] ] } }, "correctAnswer": "(−4, −3) and (3, 4)", "difficulty": 3, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² + x − 12 = 0 → (x + 4)(x − 3) = 0, so x = −4 or x = 3. Each x goes back into y = x + 1: (−4, −3) and (3, 4).", "misconceptionCode": "am-swap-pair", "options": [ "(−4, −3) and (3, 4)", "(−3, −4) and (4, 3)", "(4, 5) and (−3, −2)", "(−4, −3)" ], "correctIndex": 0 }, { "id": "sys-q14", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x − 1 and x² + y² = 25 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(−4, −3) and (3, 4)", "misconceptionDistractors": [ { "value": "(3, 2) and (−4, −5)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." }, { "value": "(−3, −4)", "code": "am-one-root", "feedback": "A quadratic from substitution usually gives TWO x-values, and each one needs its own y. Keep both pairs." } ], "scenario": { "kind": "system-substitute", "shape": "circle", "given": "y = x − 1 and x² + y² = 25", "linear": "y = x − 1", "nonlinear": "x² + y² = 25", "substituted": "x² − x − 12 = 0", "factored": "(x + 3)(x − 4) = 0", "pairs": [ "(−3, −4)", "(4, 3)" ], "pairValues": [ [ -3, -4 ], [ 4, 3 ] ] } }, "correctAnswer": "(−3, −4) and (4, 3)", "difficulty": 3, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − x − 12 = 0 → (x + 3)(x − 4) = 0, so x = −3 or x = 4. Each x goes back into y = x − 1: (−3, −4) and (4, 3).", "misconceptionCode": "am-swap-pair", "options": [ "(−4, −3) and (3, 4)", "(−3, −4) and (4, 3)", "(3, 2) and (−4, −5)", "(−3, −4)" ], "correctIndex": 1 }, { "id": "sys-q15", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x + 7 and x² + y² = 25 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(4, 11) and (3, 10)", "misconceptionDistractors": [ { "value": "(3, −4) and (4, −3)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(−4, 3)", "code": "am-one-root", "feedback": "A quadratic from substitution usually gives TWO x-values, and each one needs its own y. Keep both pairs." } ], "scenario": { "kind": "system-substitute", "shape": "circle", "given": "y = x + 7 and x² + y² = 25", "linear": "y = x + 7", "nonlinear": "x² + y² = 25", "substituted": "x² + 7x + 12 = 0", "factored": "(x + 4)(x + 3) = 0", "pairs": [ "(−4, 3)", "(−3, 4)" ], "pairValues": [ [ -4, 3 ], [ -3, 4 ] ] } }, "correctAnswer": "(−4, 3) and (−3, 4)", "difficulty": 3, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² + 7x + 12 = 0 → (x + 4)(x + 3) = 0, so x = −4 or x = −3. Each x goes back into y = x + 7: (−4, 3) and (−3, 4).", "misconceptionCode": "am-root-sign", "options": [ "(4, 11) and (3, 10)", "(3, −4) and (4, −3)", "(−4, 3) and (−3, 4)", "(−4, 3)" ], "correctIndex": 2 }, { "id": "sys-q16", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x + 10 and x² + y² = 100 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(−10, 0)", "misconceptionDistractors": [ { "value": "(0, −10) and (10, 0)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(10, 20) and (0, 10)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." } ], "scenario": { "kind": "system-substitute", "shape": "circle", "given": "y = x + 10 and x² + y² = 100", "linear": "y = x + 10", "nonlinear": "x² + y² = 100", "substituted": "x² + 10x = 0", "factored": "(x + 10)x = 0", "pairs": [ "(−10, 0)", "(0, 10)" ], "pairValues": [ [ -10, 0 ], [ 0, 10 ] ] } }, "correctAnswer": "(−10, 0) and (0, 10)", "difficulty": 4, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² + 10x = 0 → (x + 10)x = 0, so x = −10 or x = 0. Each x goes back into y = x + 10: (−10, 0) and (0, 10).", "misconceptionCode": "am-one-root", "options": [ "(0, −10) and (10, 0)", "(−10, 0)", "(10, 20) and (0, 10)", "(−10, 0) and (0, 10)" ], "correctIndex": 3 }, { "id": "sys-q17", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x − 10 and x² + y² = 100 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(0, −10)", "misconceptionDistractors": [ { "value": "(−10, 0) and (0, 10)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(0, −10) and (−10, −20)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." } ], "scenario": { "kind": "system-substitute", "shape": "circle", "given": "y = x − 10 and x² + y² = 100", "linear": "y = x − 10", "nonlinear": "x² + y² = 100", "substituted": "x² − 10x = 0", "factored": "x(x − 10) = 0", "pairs": [ "(0, −10)", "(10, 0)" ], "pairValues": [ [ 0, -10 ], [ 10, 0 ] ] } }, "correctAnswer": "(0, −10) and (10, 0)", "difficulty": 4, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − 10x = 0 → x(x − 10) = 0, so x = 0 or x = 10. Each x goes back into y = x − 10: (0, −10) and (10, 0).", "misconceptionCode": "am-one-root", "options": [ "(0, −10) and (10, 0)", "(−10, 0) and (0, 10)", "(0, −10)", "(0, −10) and (−10, −20)" ], "correctIndex": 0 }, { "id": "sys-q18", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x − 14 and x² + y² = 100 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(6, −8) and (8, −5)", "misconceptionDistractors": [ { "value": "(−8, 6) and (−6, 8)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(−6, −20) and (−8, −22)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." } ], "scenario": { "kind": "system-substitute", "shape": "circle", "given": "y = x − 14 and x² + y² = 100", "linear": "y = x − 14", "nonlinear": "x² + y² = 100", "substituted": "x² − 14x + 48 = 0", "factored": "(x − 6)(x − 8) = 0", "pairs": [ "(6, −8)", "(8, −6)" ], "pairValues": [ [ 6, -8 ], [ 8, -6 ] ] } }, "correctAnswer": "(6, −8) and (8, −6)", "difficulty": 4, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − 14x + 48 = 0 → (x − 6)(x − 8) = 0, so x = 6 or x = 8. Each x goes back into y = x − 14: (6, −8) and (8, −6).", "misconceptionCode": "am-arithmetic", "options": [ "(−8, 6) and (−6, 8)", "(6, −8) and (8, −6)", "(6, −8) and (8, −5)", "(−6, −20) and (−8, −22)" ], "correctIndex": 1 }, { "id": "sys-q19", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x + 3 and xy = 10 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(−2, −5) and (5, 2)", "misconceptionDistractors": [ { "value": "(5, 8) and (−2, 1)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." }, { "value": "(−5, −2)", "code": "am-one-root", "feedback": "A quadratic from substitution usually gives TWO x-values, and each one needs its own y. Keep both pairs." } ], "scenario": { "kind": "system-substitute", "shape": "hyperbola", "given": "y = x + 3 and xy = 10", "linear": "y = x + 3", "nonlinear": "xy = 10", "substituted": "x² + 3x − 10 = 0", "factored": "(x + 5)(x − 2) = 0", "pairs": [ "(−5, −2)", "(2, 5)" ], "pairValues": [ [ -5, -2 ], [ 2, 5 ] ] } }, "correctAnswer": "(−5, −2) and (2, 5)", "difficulty": 2, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² + 3x − 10 = 0 → (x + 5)(x − 2) = 0, so x = −5 or x = 2. Each x goes back into y = x + 3: (−5, −2) and (2, 5).", "misconceptionCode": "am-swap-pair", "options": [ "(5, 8) and (−2, 1)", "(−5, −2)", "(−5, −2) and (2, 5)", "(−2, −5) and (5, 2)" ], "correctIndex": 2 }, { "id": "sys-q20", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x − 5 and xy = −4 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(−1, −6) and (−4, −9)", "misconceptionDistractors": [ { "value": "(−4, 1) and (−1, 4)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(1, −4)", "code": "am-one-root", "feedback": "A quadratic from substitution usually gives TWO x-values, and each one needs its own y. Keep both pairs." } ], "scenario": { "kind": "system-substitute", "shape": "hyperbola", "given": "y = x − 5 and xy = −4", "linear": "y = x − 5", "nonlinear": "xy = −4", "substituted": "x² − 5x + 4 = 0", "factored": "(x − 1)(x − 4) = 0", "pairs": [ "(1, −4)", "(4, −1)" ], "pairValues": [ [ 1, -4 ], [ 4, -1 ] ] } }, "correctAnswer": "(1, −4) and (4, −1)", "difficulty": 2, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − 5x + 4 = 0 → (x − 1)(x − 4) = 0, so x = 1 or x = 4. Each x goes back into y = x − 5: (1, −4) and (4, −1).", "misconceptionCode": "am-root-sign", "options": [ "(−4, 1) and (−1, 4)", "(1, −4)", "(−1, −6) and (−4, −9)", "(1, −4) and (4, −1)" ], "correctIndex": 3 }, { "id": "sys-q21", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x + 5 and xy = −6 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(−3, −2) and (−2, −3)", "misconceptionDistractors": [ { "value": "(2, −3) and (3, −2)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(3, 8) and (2, 7)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." } ], "scenario": { "kind": "system-substitute", "shape": "hyperbola", "given": "y = x + 5 and xy = −6", "linear": "y = x + 5", "nonlinear": "xy = −6", "substituted": "x² + 5x + 6 = 0", "factored": "(x + 3)(x + 2) = 0", "pairs": [ "(−3, 2)", "(−2, 3)" ], "pairValues": [ [ -3, 2 ], [ -2, 3 ] ] } }, "correctAnswer": "(−3, 2) and (−2, 3)", "difficulty": 3, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² + 5x + 6 = 0 → (x + 3)(x + 2) = 0, so x = −3 or x = −2. Each x goes back into y = x + 5: (−3, 2) and (−2, 3).", "misconceptionCode": "am-sign-pair", "options": [ "(−3, 2) and (−2, 3)", "(−3, −2) and (−2, −3)", "(2, −3) and (3, −2)", "(3, 8) and (2, 7)" ], "correctIndex": 0 }, { "id": "sys-q22", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x − 2 and xy = 3 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(1, −1) and (−3, −5)", "misconceptionDistractors": [ { "value": "(−3, −1) and (1, 3)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(−1, −3)", "code": "am-one-root", "feedback": "A quadratic from substitution usually gives TWO x-values, and each one needs its own y. Keep both pairs." } ], "scenario": { "kind": "system-substitute", "shape": "hyperbola", "given": "y = x − 2 and xy = 3", "linear": "y = x − 2", "nonlinear": "xy = 3", "substituted": "x² − 2x − 3 = 0", "factored": "(x + 1)(x − 3) = 0", "pairs": [ "(−1, −3)", "(3, 1)" ], "pairValues": [ [ -1, -3 ], [ 3, 1 ] ] } }, "correctAnswer": "(−1, −3) and (3, 1)", "difficulty": 3, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − 2x − 3 = 0 → (x + 1)(x − 3) = 0, so x = −1 or x = 3. Each x goes back into y = x − 2: (−1, −3) and (3, 1).", "misconceptionCode": "am-root-sign", "options": [ "(1, −1) and (−3, −5)", "(−1, −3) and (3, 1)", "(−3, −1) and (1, 3)", "(−1, −3)" ], "correctIndex": 1 }, { "id": "sys-q23", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x − 7 and xy = −6 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(1, −6)", "misconceptionDistractors": [ { "value": "(−6, 1) and (−1, 6)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(−1, −8) and (−6, −13)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." } ], "scenario": { "kind": "system-substitute", "shape": "hyperbola", "given": "y = x − 7 and xy = −6", "linear": "y = x − 7", "nonlinear": "xy = −6", "substituted": "x² − 7x + 6 = 0", "factored": "(x − 1)(x − 6) = 0", "pairs": [ "(1, −6)", "(6, −1)" ], "pairValues": [ [ 1, -6 ], [ 6, -1 ] ] } }, "correctAnswer": "(1, −6) and (6, −1)", "difficulty": 3, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² − 7x + 6 = 0 → (x − 1)(x − 6) = 0, so x = 1 or x = 6. Each x goes back into y = x − 7: (1, −6) and (6, −1).", "misconceptionCode": "am-one-root", "options": [ "(1, −6)", "(−6, 1) and (−1, 6)", "(1, −6) and (6, −1)", "(−1, −8) and (−6, −13)" ], "correctIndex": 2 }, { "id": "sys-q24", "objectiveRecordIds": [ "0580-extended-E2.5-5" ], "prompt": "Solve y = x + 4 and xy = −3 together.", "data": { "family": "system-substitute", "misconceptionDistractor": "(−3, 1) and (−1, 4)", "misconceptionDistractors": [ { "value": "(1, −3) and (3, −1)", "code": "am-swap-pair", "feedback": "Back-substitute each x into the LINEAR equation: the y that goes with x = 3 is only 3 when the line says y = x." }, { "value": "(3, 7) and (1, 5)", "code": "am-root-sign", "feedback": "A bracket (x − 3) gives the root x = 3, not x = −3: the root is the value that makes the bracket ZERO." } ], "scenario": { "kind": "system-substitute", "shape": "hyperbola", "given": "y = x + 4 and xy = −3", "linear": "y = x + 4", "nonlinear": "xy = −3", "substituted": "x² + 4x + 3 = 0", "factored": "(x + 3)(x + 1) = 0", "pairs": [ "(−3, 1)", "(−1, 3)" ], "pairValues": [ [ -3, 1 ], [ -1, 3 ] ] } }, "correctAnswer": "(−3, 1) and (−1, 3)", "difficulty": 4, "hint": "Substitute the WHOLE linear expression for y — in brackets — then multiply it out and simplify to a quadratic equal to zero. Factorise it, and put EACH x back into the linear equation to get the y that goes with it.", "explanation": "Replace y with the linear expression: x² + 4x + 3 = 0 → (x + 3)(x + 1) = 0, so x = −3 or x = −1. Each x goes back into y = x + 4: (−3, 1) and (−1, 3).", "misconceptionCode": "am-arithmetic", "options": [ "(1, −3) and (3, −1)", "(−3, 1) and (−1, 4)", "(3, 7) and (1, 5)", "(−3, 1) and (−1, 3)" ], "correctIndex": 3 } ] } } }