Significant Figures and Estimation
Stage 8 of 23 Strand 3 of 8 5 lessons
5 illustrated lessons, each teaching the why before the how.
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Significant Figures #
Counting digits from the leading nonzero one.
Significant figures are counted from the leading nonzero digit
A crowd of 2347 is reported as about 2000 — rounded to one significant figure.
Keep one figure and let the digit after it decide: 3 is under 5, so 2000.
Two significant figures keeps 23 and rounds on the 4: 2347 becomes 2300.
Counting starts at the 5, the first digit that is not zero — so 0.0052 has two.
Now you
Round 0.026 to one significant figure
How many significant figures does 0.0085 have?
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Bounds and Error Intervals #
Half a unit of doubt, open at the top.
A rounded measurement pins the truth inside a half-unit band that is open at the top
"12 cm, to the nearest cm" means the true length lives between 11.5 and 12.5.
The band is always half the rounding unit each side, whatever the unit is.
The top stays out: 12.5 would round up to 13, so its circle is hollow — x < 12.5.
Bounds travel through a sum: worst cases add, upper with upper, lower with lower.
Now you
A mass is 11 kg to the nearest kg. Which value could it not be?
Rods measure 9 cm and 4 cm, each to the nearest cm. Largest possible total?
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Propagating Upper and Lower Bounds #
Each bound chosen to push the answer up, or down.
To make a result as large as it can be, pick each measurement's bound that pushes it up
For a product, the upper bound multiplies the two upper bounds, and the lower bound the two lower bounds.
A difference is largest when the number you start from is as big as it can be and the number taken away is as small as it can be: upper minus lower.
A quotient is largest with the biggest top and the smallest bottom: upper bound divided by lower bound.
Give the answer to the most accurate level at which both bounds round to the same value.
Now you
a = 15 and b = 9, each to the nearest whole number. What is the upper bound of a − b?
A rectangle measures 9 cm by 8 cm, each to the nearest cm. What is the upper bound of its area, in cm²?
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Percentage Error #
The gap, measured against the exact value.
Percentage error measures an error against the exact value it came from
A rod is exactly 200 cm and a tape reads 194 cm. The error is 6 cm.
Divide the error by the exact value, then turn the answer into a percentage.
The same 6 cm off a 20 cm rod is 6 ÷ 20 = 30% — ten times the error, in effect.
One rule: the size of the gap over the exact value, written as a percentage.
Now you
A length is exactly 400 mm and the measurement reads 480 mm. What is the percentage error?
Two scales are each 5 g out. One weighs 50 g, the other 500 g. Which has the larger percentage error?
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Estimating with One Significant Figure #
Round everything first; the size comes out right.
Round each number to one significant figure to know the size of an answer in advance
How big is 3.8 × 52? Round each to one significant figure: 4 × 50.
4 × 50 is 200, and the exact answer — 197.6 — lives right beside it.
Decimals round the same way: 0.52 × 88 is close to 0.5 × 90, which is 45.
A claimed answer of 40 for 3.8 × 52 fails the size check — the estimate says 200.
Now you
Roughly, 2.3 × 33 is…
Roughly, 3.2 × 89 is…
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