bol $<$ is used at the top because 21.5 would round to 22, not 21. In bounds questions, write the interval first and then use it in the calculation."}]},{"id":"card-2","type":"mcq","order":2,"options":[{"id":"a","text":"$$20.5 \\le x < 21.5$","isCorrect":true},{"id":"b","text":"$$20 < x < 22$","isCorrect":false},{"id":"c","text":"$$20.5 < x \\le 21.5$","isCorrect":false},{"id":"d","text":"$$21 \\le x < 22$","isCorrect":false}],"question":"A rod has length 21 cm to the nearest cm. Which interval is correct for its true length $x$?","explanation":"$$21$ to the nearest cm means the rounding unit is 1, so the half-unit is 0.5 and the interval is $20.5 \\le x < 21.5$. The most common wrong answer is C because students reverse the inclusive and exclusive ends, but 20.5 does round to 21 so it must be included, while 21.5 rounds to 22 so it must be excluded.","correctOptionId":"a"},{"id":"card-3","type":"content","order":3,"title":"The half-unit rule","blocks":[{"id":"block-1","content":"To find bounds, take half of the rounding unit."},{"id":"block-2","content":"Then subtract that half-unit to get the lower bound and add it to get the upper bound. Rounding to the nearest cm gives a half-unit of 0.5 cm, rounding to the nearest 5 m gives 2.5 m, and rounding to 2 decimal places gives 0.005."},{"id":"block-3","content":"Example: if a length is 130 m to the nearest 5 m, the half-unit is 2.5 m. Then $130 - 2.5 = 127.5$ and $130 + 2.5 = 132.5$. So the actual length is from 127.5 m up to but not including 132.5 m."},{"id":"block-4","content":"For significant figures, first find the place value of the last significant digit. If 3680 is correct to 3 significant figures, the last significant digit is in the tens place, so it was rounded to the nearest 10 and the half-unit is 5. Then $3680 - 5 = 3675$ and $3680 + 5 = 3685$. So the actual value is from 3675 up to but not including 3685."}]},{"id":"card-4","type":"fill_blank","order":4,"blanks":[{"id":"lower","isMath":false,"correctAnswer":"7.815"},{"id":"upper","isMath":false,"correctAnswer":"7.825"}],"sentence":"A mass is 7.82 kg correct to 2 decimal places. The lower bound is {{blank:lower}} kg and the upper bound is {{blank:upper}} kg."},{"id":"card-5","type":"flashcard","cards":[{"id":"fc-1","back":"The smallest possible value that rounds to the stated measurement.","front":"Lower bound"},{"id":"fc-2","back":"The top boundary of the interval, usually not included.","front":"Upper bound"},{"id":"fc-3","back":"A way to show all possible values, such as $L \\le x < U$.","front":"Interval notation"},{"id":"fc-4","back":"Find half of the rounding unit, then subtract and add it to the rounded value.","front":"Half-unit rule"}],"order":5},{"id":"card-6","type":"content","order":6,"title":"Bounds in perimeter and area","blocks":[{"id":"block-1","content":"For positive lengths, making a measurement bigger makes perimeter, area, and circumference bigger. So upper bound questions use bigger lengths, and lower bound questions use smaller lengths."},{"id":"block-2","content":"A rectangle has length 38 cm to the nearest cm and width 7.4 cm to the nearest mm. Therefore $37.5 \\le L < 38.5$ and $7.35 \\le W < 7.45$ because nearest mm is nearest 0.1 cm."},{"id":"block-3","content":"The upper bound of the perimeter uses both upper values because perimeter is found by adding positive lengths. $$P_{\\text{upper}} = 2(38.5 + 7.45) = 91.9$$ So the upper bound of the perimeter is 91.9 cm."},{"id":"block-4","content":"The lower bound of the area uses both lower values because we want the smallest possible product. $$A_{\\text{lower}} = 37.5 \\times 7.35 = 275.625$$ So the lower bound of the area is 275.625 $\\text{cm}^2$."}]},{"id":"card-7","type":"mcq","order":7,"options":[{"id":"a","text":"$$2(12.5 + 4.65)$","isCorrect":true},{"id":"b","text":"$$2(11.5 + 4.55)$","isCorrect":false},{"id":"c","text":"$$2(12.5 + 4.55)$","isCorrect":false},{"id":"d","text":"$$2(11.5 + 4.65)$","isCorrect":false}],"question":"A rectangle has length 12 cm to the nearest cm and width 4.6 cm to the nearest mm. Which calculation gives

Bounds Lesson