{
  "schemaVersion": "1.0",
  "moduleId": "bounds-estimation-0580-pilot",
  "status": "prototype-scenarios",
  "pathways": {
    "accuracy-and-bounds": {
      "objectiveCode": "C1.10",
      "learnerAction": "Move the reading, the accuracy dial and the two ends of the band until the bench shows what the goal states.",
      "objectiveVerbs": [
        "treat a rounded value as the band of values it really is",
        "give the upper and lower bounds of a rounded measurement",
        "find the bounds of a result calculated from rounded data"
      ],
      "responseType": "model-construction",
      "scenarios": [
        {
          "id": "bounds-mission-01",
          "title": "Show every length that reads as 4.6 cm",
          "objectiveRecordIds": [
            "0580-core-C1.10-section",
            "0580-extended-E1.10-1"
          ],
          "prompt": "Show every length that reads as 4.6 cm. Goal: the tape was read to the nearest 0.1 cm · the band starts at the shortest length that still reads as the tape · the band ends just below the length that would read as the next tenth.",
          "model": {
            "start": {
              "kind": "bench",
              "meas": "tape",
              "reading": 4.6,
              "step": 0.1,
              "lb": 4.5,
              "ub": 4.7,
              "a": null,
              "aStep": null,
              "b": null,
              "bStep": null,
              "want": null
            },
            "editable": [
              "lb",
              "ub"
            ],
            "given": [
              "reading",
              "step"
            ],
            "object": "strip of tape"
          },
          "expected": {
            "accuracy-is": 0.1,
            "lb-is": 4.55,
            "ub-is": 4.65
          },
          "goal": {
            "text": "the tape was read to the nearest 0.1 cm · the band starts at the shortest length that still reads as the tape · the band ends just below the length that would read as the next tenth",
            "statements": [
              {
                "kind": "accuracy-is",
                "text": "the tape was read to the nearest 0.1 cm",
                "code": "accuracy-is-half-width",
                "value": 0.1
              },
              {
                "kind": "lb-is",
                "text": "the band starts at the shortest length that still reads as the tape",
                "code": "bound-whole-step",
                "value": 4.55
              },
              {
                "kind": "ub-is",
                "text": "the band ends just below the length that would read as the next tenth",
                "code": "bound-whole-step",
                "value": 4.65
              }
            ],
            "working": [
              "The accuracy is 0.1 cm, so each end sits 0.05 cm from the reading, not 0.1 cm.",
              "Drag the left end to 4.55 cm and the right end to 4.65 cm: 4.65 cm itself would read as 4.7 cm, so the band stops just short of it."
            ]
          },
          "solution": {
            "kind": "bench",
            "meas": "tape",
            "reading": 4.6,
            "step": 0.1,
            "lb": 4.55,
            "ub": 4.65,
            "a": null,
            "aStep": null,
            "b": null,
            "bStep": null,
            "want": null
          },
          "hint": "The band is twice as wide as it should be: each end must be only half the accuracy from the reading.",
          "misconceptionCode": "bound-whole-step"
        },
        {
          "id": "bounds-mission-02",
          "title": "Make the band for a rod read to the nearest 0.5 cm",
          "objectiveRecordIds": [
            "0580-core-C1.10-section",
            "0580-extended-E1.10-1"
          ],
          "prompt": "Make the band for a rod read to the nearest 0.5 cm. Goal: the rod was read to the nearest 0.5 cm · the reading sits in the middle of the band · the shortest length that still reads as the rod is inside the band · the length that would read as the next half cm up is outside the band.",
          "model": {
            "start": {
              "kind": "bench",
              "meas": "rod",
              "reading": 3,
              "step": 0.1,
              "lb": 2.95,
              "ub": 3.05,
              "a": null,
              "aStep": null,
              "b": null,
              "bStep": null,
              "want": null
            },
            "editable": [
              "step",
              "lb",
              "ub"
            ],
            "given": [
              "reading"
            ],
            "object": "metal rod"
          },
          "expected": {
            "accuracy-is": 0.5,
            "band-centred-on-reading": true,
            "probe-inside": 2.75,
            "probe-outside": 3.25
          },
          "goal": {
            "text": "the rod was read to the nearest 0.5 cm · the reading sits in the middle of the band · the shortest length that still reads as the rod is inside the band · the length that would read as the next half cm up is outside the band",
            "statements": [
              {
                "kind": "accuracy-is",
                "text": "the rod was read to the nearest 0.5 cm",
                "code": "accuracy-is-half-width",
                "value": 0.5
              },
              {
                "kind": "band-centred-on-reading",
                "text": "the reading sits in the middle of the band",
                "code": "bound-whole-step",
                "value": true
              },
              {
                "kind": "probe-inside",
                "text": "the shortest length that still reads as the rod is inside the band",
                "code": "upper-end-included",
                "value": 2.75
              },
              {
                "kind": "probe-outside",
                "text": "the length that would read as the next half cm up is outside the band",
                "code": "upper-end-included",
                "value": 3.25
              }
            ],
            "working": [
              "Set the accuracy dial first: it decides how wide the band can be.",
              "Then test values at the ends: 2.75 cm rounds to 3 cm, so it belongs inside, while 3.25 cm rounds to 3.5 cm, so it does not."
            ]
          },
          "solution": {
            "kind": "bench",
            "meas": "rod",
            "reading": 3,
            "step": 0.5,
            "lb": 2.75,
            "ub": 3.25,
            "a": null,
            "aStep": null,
            "b": null,
            "bStep": null,
            "want": null
          },
          "hint": "Change the accuracy dial, then move both ends until each one is half an accuracy from the reading.",
          "misconceptionCode": "accuracy-is-half-width"
        },
        {
          "id": "bounds-mission-03",
          "title": "A band 0.2 cm wide with the reading in its middle",
          "objectiveRecordIds": [
            "0580-core-C1.10-section",
            "0580-extended-E1.10-1"
          ],
          "prompt": "A band 0.2 cm wide with the reading in its middle. Goal: the band is 0.2 cm wide from end to end · the pencil reads 12.4 cm · the reading sits in the middle of the band · the band starts half a width below the reading.",
          "model": {
            "start": {
              "kind": "bench",
              "meas": "pencil",
              "reading": 12.4,
              "step": 0.2,
              "lb": 12.2,
              "ub": 12.6,
              "a": null,
              "aStep": null,
              "b": null,
              "bStep": null,
              "want": null
            },
            "editable": [
              "reading",
              "lb",
              "ub"
            ],
            "given": [
              "step"
            ],
            "object": "pencil"
          },
          "expected": {
            "band-width-is": 0.2,
            "reading-is": 12.4,
            "band-centred-on-reading": true,
            "lb-is": 12.3
          },
          "goal": {
            "text": "the band is 0.2 cm wide from end to end · the pencil reads 12.4 cm · the reading sits in the middle of the band · the band starts half a width below the reading",
            "statements": [
              {
                "kind": "band-width-is",
                "text": "the band is 0.2 cm wide from end to end",
                "code": "width-is-half-step",
                "value": 0.2
              },
              {
                "kind": "reading-is",
                "text": "the pencil reads 12.4 cm",
                "code": "bound-whole-step",
                "value": 12.4
              },
              {
                "kind": "band-centred-on-reading",
                "text": "the reading sits in the middle of the band",
                "code": "step-is-reading",
                "value": true
              },
              {
                "kind": "lb-is",
                "text": "the band starts half a width below the reading",
                "code": "bound-whole-step",
                "value": 12.3
              }
            ],
            "working": [
              "A band 0.2 cm wide reaches 0.1 cm each side of the reading, so the accuracy is the nearest 0.2 cm.",
              "Keep the reading at 12.4 cm and move both ends until they are 12.3 cm and 12.5 cm."
            ]
          },
          "solution": {
            "kind": "bench",
            "meas": "pencil",
            "reading": 12.4,
            "step": 0.2,
            "lb": 12.3,
            "ub": 12.5,
            "a": null,
            "aStep": null,
            "b": null,
            "bStep": null,
            "want": null
          },
          "hint": "Half the width goes each side: take 0.1 cm from the reading for one end and add it for the other.",
          "misconceptionCode": "width-is-half-step"
        },
        {
          "id": "bounds-mission-04",
          "title": "The greatest and least perimeter of the rounded rectangle",
          "objectiveRecordIds": [
            "0580-extended-E1.10-1",
            "0580-extended-E1.10-2"
          ],
          "prompt": "The greatest and least perimeter of the rounded rectangle. Goal: the band starts at the least perimeter the two sides allow · the band ends at the greatest perimeter the two sides allow.",
          "model": {
            "start": {
              "kind": "result",
              "meas": "rect",
              "reading": null,
              "step": null,
              "lb": 20,
              "ub": 20,
              "a": 7.0,
              "aStep": 1.0,
              "b": 3.0,
              "bStep": 1.0,
              "want": "perimeter"
            },
            "editable": [
              "lb",
              "ub"
            ],
            "given": [
              "a",
              "aStep",
              "b",
              "bStep",
              "want"
            ],
            "object": "rectangle drawn on a grid"
          },
          "expected": {
            "result-lb-is": 18.0,
            "result-ub-is": 22.0
          },
          "goal": {
            "text": "the band starts at the least perimeter the two sides allow · the band ends at the greatest perimeter the two sides allow",
            "statements": [
              {
                "kind": "result-lb-is",
                "text": "the band starts at the least perimeter the two sides allow",
                "code": "result-used-readings",
                "value": 18.0
              },
              {
                "kind": "result-ub-is",
                "text": "the band ends at the greatest perimeter the two sides allow",
                "code": "result-used-readings",
                "value": 22.0
              }
            ],
            "working": [
              "Each side is its own band: 7 cm means at least 6.5 cm and just under 7.5 cm, and 3 cm means at least 2.5 cm and just under 3.5 cm.",
              "The perimeter uses both sides twice, so its least is 2 × (6.5 cm + 2.5 cm) = 18 cm and its greatest is 2 × (7.5 cm + 3.5 cm) = 22 cm."
            ]
          },
          "solution": {
            "kind": "result",
            "meas": "rect",
            "reading": null,
            "step": null,
            "lb": 18.0,
            "ub": 22.0,
            "a": 7.0,
            "aStep": 1.0,
            "b": 3.0,
            "bStep": 1.0,
            "want": "perimeter"
          },
          "hint": "Do not double the readings: put the smallest sides in for the left end and the biggest for the right.",
          "misconceptionCode": "result-used-readings"
        },
        {
          "id": "bounds-mission-05",
          "title": "Show the band for 3.4 cm read to 2 significant figures",
          "objectiveRecordIds": [
            "0580-core-C1.10-section",
            "0580-extended-E1.10-1"
          ],
          "prompt": "Show the band for 3.4 cm read to 2 significant figures. Goal: the caterpillar was read to 2 significant figures · the band starts at the shortest length that still reads as the caterpillar · the band ends just below the length that would read longer · the reading sits in the middle of the band.",
          "model": {
            "start": {
              "kind": "bench",
              "meas": "worm",
              "reading": 3.4,
              "step": 1,
              "lb": 2.9,
              "ub": 3.9,
              "a": null,
              "aStep": null,
              "b": null,
              "bStep": null,
              "want": null
            },
            "editable": [
              "reading",
              "step",
              "lb",
              "ub"
            ],
            "given": [],
            "object": "caterpillar"
          },
          "expected": {
            "accuracy-is": 0.1,
            "lb-is": 3.35,
            "ub-is": 3.45,
            "band-centred-on-reading": true
          },
          "goal": {
            "text": "the caterpillar was read to 2 significant figures · the band starts at the shortest length that still reads as the caterpillar · the band ends just below the length that would read longer · the reading sits in the middle of the band",
            "statements": [
              {
                "kind": "accuracy-is",
                "text": "the caterpillar was read to 2 significant figures",
                "code": "accuracy-is-half-width",
                "value": 0.1
              },
              {
                "kind": "lb-is",
                "text": "the band starts at the shortest length that still reads as the caterpillar",
                "code": "bound-whole-step",
                "value": 3.35
              },
              {
                "kind": "ub-is",
                "text": "the band ends just below the length that would read longer",
                "code": "bound-whole-step",
                "value": 3.45
              },
              {
                "kind": "band-centred-on-reading",
                "text": "the reading sits in the middle of the band",
                "code": "step-is-reading",
                "value": true
              }
            ],
            "working": [
              "To 2 significant figures the last digit of 3.4 cm is a tenth, so the accuracy is 0.1 cm, not 1 cm.",
              "Half of 0.1 cm is 0.05 cm, so the band runs from 3.35 cm to 3.45 cm with 3.4 cm in the middle."
            ]
          },
          "solution": {
            "kind": "bench",
            "meas": "worm",
            "reading": 3.4,
            "step": 0.1,
            "lb": 3.35,
            "ub": 3.45,
            "a": null,
            "aStep": null,
            "b": null,
            "bStep": null,
            "want": null
          },
          "hint": "Set the accuracy dial first, then move both ends in to half of it either side of the reading.",
          "misconceptionCode": "accuracy-is-half-width"
        },
        {
          "id": "bounds-mission-06",
          "title": "The greatest and least area of the same rounded rectangle",
          "objectiveRecordIds": [
            "0580-extended-E1.10-1",
            "0580-extended-E1.10-2"
          ],
          "prompt": "The greatest and least area of the same rounded rectangle. Goal: the band starts at the least area the two sides allow · the band ends at the greatest area the two sides allow.",
          "model": {
            "start": {
              "kind": "result",
              "meas": "rect",
              "reading": null,
              "step": null,
              "lb": 21.0,
              "ub": 21.0,
              "a": 7.0,
              "aStep": 1.0,
              "b": 3.0,
              "bStep": 1.0,
              "want": "area"
            },
            "editable": [
              "lb",
              "ub"
            ],
            "given": [
              "a",
              "aStep",
              "b",
              "bStep",
              "want"
            ],
            "object": "rectangle drawn on a grid"
          },
          "expected": {
            "result-lb-is": 16.25,
            "result-ub-is": 26.25
          },
          "goal": {
            "text": "the band starts at the least area the two sides allow · the band ends at the greatest area the two sides allow",
            "statements": [
              {
                "kind": "result-lb-is",
                "text": "the band starts at the least area the two sides allow",
                "code": "result-used-readings",
                "value": 16.25
              },
              {
                "kind": "result-ub-is",
                "text": "the band ends at the greatest area the two sides allow",
                "code": "result-used-readings",
                "value": 26.25
              }
            ],
            "working": [
              "The two sides are two bands: 7 cm means at least 6.5 cm and just under 7.5 cm, and 3 cm means at least 2.5 cm and just under 3.5 cm.",
              "Area multiplies the sides, so the least is 6.5 cm × 2.5 cm = 16.25 cm² and the greatest is 7.5 cm × 3.5 cm = 26.25 cm². The rectangle at the readings, 21 cm², sits inside that band but is neither end of it."
            ]
          },
          "solution": {
            "kind": "result",
            "meas": "rect",
            "reading": null,
            "step": null,
            "lb": 16.25,
            "ub": 26.25,
            "a": 7.0,
            "aStep": 1.0,
            "b": 3.0,
            "bStep": 1.0,
            "want": "area"
          },
          "hint": "The area at the readings is not an end of the band: multiply the two smallest ends together for the left of your band, and the two biggest for the right.",
          "misconceptionCode": "result-used-readings"
        }
      ]
    }
  }
}