nearest cm, the interval is $27.5 \\le l < 28.5$."},{"id":"block-3","content":"To find bounds, work out the rounding gap and halve it. Nearest cm gives a half-gap of $0.5$, nearest 5 m gives $2.5$, and correct to 2 decimal places gives $0.005$."}]},{"id":"card-2","type":"mcq","order":2,"options":[{"id":"a","text":"$$27.5 \\le l < 28.5$","isCorrect":true},{"id":"b","text":"$$27 < l < 29$","isCorrect":false},{"id":"c","text":"$$27.5 < l \\le 28.5$","isCorrect":false},{"id":"d","text":"$$28 \\le l < 29$","isCorrect":false}],"question":"A length is 28 cm to the nearest cm. Which interval is correct?","explanation":"A is correct because the half-gap is $0.5$, so the true length is from $27.5$ up to but not including $28.5$. The most common wrong answer is C, but $27.5$ does round to 28 and should be included, while $28.5$ rounds up to 29 so it must be excluded. Hint: Halve the rounding interval, then check which end is excluded.","correctOptionId":"a"},{"id":"card-3","type":"content","order":3,"title":"Working from the rounding clue","blocks":[{"id":"block-1","content":"A measurement of 7.6 cm correct to the nearest mm is rounded to the nearest $0.1$ cm, because 1 mm is $0.1$ cm. So the half-gap is $0.05$ cm."},{"id":"block-2","content":"That gives $7.55 \\le w < 7.65$. A common mistake is to use $0.5$ cm, which is the half-gap for nearest cm, not nearest mm."},{"id":"block-3","content":"For 58.0 cm correct to the nearest mm, the bounds are $57.95 \\le l < 58.05$. For a number correct to 2 decimal places, the half-gap is $0.005$."}]},{"id":"card-4","type":"fill_blank","order":4,"blanks":[{"id":"lower","isMath":true,"correctAnswer":"57.95"},{"id":"upper","isMath":true,"correctAnswer":"58.05"}],"sentence":"For 58.0 cm correct to the nearest mm, the lower bound is {{blank:lower}} cm and the upper bound is {{blank:upper}} cm."},{"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 largest boundary value before the rounded value changes.","front":"Upper bound"},{"id":"fc-3","back":"Half of the rounding interval, added and subtracted to find bounds.","front":"Half-gap"},{"id":"fc-4","back":"A way to write all possible values, for example $27.5 \\le l < 28.5$.","front":"Interval notation"}],"order":5},{"id":"card-6","type":"content","order":6,"title":"Bounds in geometry","blocks":[{"id":"block-1","content":"When all measurements are positive, addition and multiplication follow a simple pattern. Use lower values to make a perimeter or area smaller, and upper values to make them larger."},{"id":"block-2","content":"For a rectangle with $l=19$ cm to the nearest cm and $w=6.4$ cm to the nearest mm, first write $18.5 \\le l < 19.5$ and $6.35 \\le w < 6.45$. Then $P_{\\text{upper}}=2(19.5+6.45)=51.9 \\, \\text{cm}$ and $A_{\\text{lower}}=18.5 \\times 6.35=117.475 \\, \\text{cm}^2$."},{"id":"block-3","content":"For circles, the radius controls everything: $C=2\\pi r$ and $A=\\pi r^2$. If $r=7.2$ cm correct to 1 d.p., then $7.15 \\le r < 7.25$, so $C_{\\text{lower}}=14.3\\pi$ and $A_{\\text{upper}}=52.5625\\pi$."}]},{"id":"card-7","type":"mcq","order":7,"options":[{"id":"a","text":"$$\\pi(7.25)^2$","isCorrect":true},{"id":"b","text":"$$\\pi(7.15)^2$","isCorrect":false},{"id":"c","text":"$$2\\pi(7.25)$","isCorrect":false},{"id":"d","text":"$$\\pi(7.2)^2$","isCorrect":false}],"question":"A circle has radius 7.2 cm correct to 1 decimal place. Which expression gives the upper bound for its area?","explanation":"A is correct because the upper bound of the radius is $7.25$ and area uses $A=\\pi r^2$. The most common wrong answer is D, but $7.2$ is only the rounded value, not the largest possible true radius.","correctOptionId":"a"},{"id":"card-8","type":"ordering","items":[{"id":"item-1","content":"Underline the rounding clue, such as nearest cm or 2 decimal places."},{"id":"item-2","content":"Find the half-gap and write the lower and upper bounds for each measurement."},{"id":"item-3","content":"Decide which combination

Bounds Lesson

Bounds Lesson