Bounds & Rounding flashcards
9 cards on Bounds & Rounding: the key points, worked examples and the slips to watch. Swipe through them, or print them.
Card 1 of 9
Bounds & Rounding
Decimal places and significant figures
What is the difference between rounding to decimal places and rounding to significant figures?
Answer it in your head, then turn the card over.
Decimal places and significant figures
Key points
- Decimal places count the digits after the decimal point.
- Significant figures count from the first digit that is not zero.
- Look at the next digit along: 5 or more rounds up, 4 or less leaves it.
Round 0.030462
To 3 d.p.: stop at the third digit after the point: 0.030
To 3 s.f.: start at the 3, so 3, 0, 4, and the 6 rounds up: 0.0305
Watch out
Zeros at the front never count as significant figures: they only fix the place value.
Bounds & Rounding
Rounding big numbers and keeping zeros
When you round, which zeros must you keep, and why?
Answer it in your head, then turn the card over.
Rounding big numbers and keeping zeros
Key points
- Rounding a big number: fill the places up to the decimal point with zeros.
- Those zeros keep the number its size: 36 750 to 2 s.f. is 37 000, not 37.
- Keep a final 0 after the point when it is a place you rounded to: 7.3958 to 2 d.p. is 7.40.
Round 48 260 to 2 s.f.
The first two significant figures are 4 and 8.
The next digit is 2, so round down: 48 000
Watch out
Dropping the zeros changes the number: 48 260 to 2 s.f. is 48 000, not 48.
Bounds & Rounding
Estimating a calculation
How do you estimate the answer to a calculation?
Answer it in your head, then turn the card over.
Estimating a calculation
Key points
- Round every number to 1 significant figure first.
- Write the rounded numbers down, then work it out in your head: that working earns the marks.
- Use an estimate to check that a calculator answer is sensible.
Estimate 3.92 × 58.7 ÷ 0.21
Round each number: 4 × 60 ÷ 0.2
4 × 60 = 240
240 ÷ 0.2 = 1200 (dividing by 0.2 is the same as multiplying by 5)
Watch out
Using the exact numbers, or the nearest whole number: an estimate rounds each one to 1 s.f.
Bounds & Rounding
Error intervals for rounded values
How do you write the error interval of a value that has been rounded?
Answer it in your head, then turn the card over.
Error intervals for rounded values
Key points
- Halve the unit it was rounded to, and go that far below and above the value.
- Use ≤ at the lower end and < at the upper end.
- The upper end is left out because it would itself round up.
Example
12.6 cm to 1 d.p.: half of 0.1 is 0.05, so 12.55 ≤ x < 12.65
70 to the nearest 10: half of 10 is 5, so 65 ≤ y < 75
Watch out
Writing ≤ at both ends: 12.65 rounds up to 12.7, so the upper end needs <.
Bounds & Rounding
Error intervals for truncated values
How is the error interval of a truncated value different from a rounded one?
Answer it in your head, then turn the card over.
Error intervals for truncated values
Key points
- Truncating chops off the extra digits without rounding.
- So the value can be anything from the number shown up to, but not including, the next step up.
- The interval starts at the number shown, not half a unit below it.
Example
3.62 truncated to 2 d.p.: 3.62 ≤ x < 3.63
9 truncated to a whole number: 9 ≤ x < 10
Watch out
Treating it like rounding: 5.6 truncated is 5.6 ≤ x < 5.7, not 5.55 ≤ x < 5.65.
Bounds & Rounding
Upper and lower bounds
How do you find the upper and lower bounds of a rounded measurement?
Answer it in your head, then turn the card over.
Upper and lower bounds
Key points
- The bounds sit half a unit of rounding either side of the value.
- Lower bound: the value minus half a unit. Upper bound: the value plus half a unit.
- The upper bound would itself round up, but it is still the one used in calculations.
Example
2.7 m to the nearest 0.1 m: 2.65 m and 2.75 m
600 g to the nearest 50 g: half of 50 is 25, so 575 g and 625 g
Watch out
Going a whole unit either side: 340 g to the nearest 10 g is 335 g to 345 g, not 330 g to 350 g.
Bounds & Rounding
HigherBounds when adding or multiplying
Which bounds give the largest and smallest possible answer to a + b or a × b?
Answer it in your head, then turn the card over.
Bounds when adding or multiplying
Key points
- Largest answer: use the upper bound of both numbers.
- Smallest answer: use the lower bound of both numbers.
- Write down which bound you used for each quantity: that reasoning earns marks.
8 cm by 5 cm, each to the nearest cm
Largest area: 8.5 × 5.5 = 46.75 cm²
Smallest area: 7.5 × 4.5 = 33.75 cm²
Watch out
Using the rounded values gives 40 cm², which is neither the largest nor the smallest area.
Bounds & Rounding
HigherBounds when subtracting or dividing
Which bounds give the largest possible answer to a − b or a ÷ b?
Answer it in your head, then turn the card over.
Bounds when subtracting or dividing
Key points
- Largest a − b: the biggest a, and take away the smallest b.
- Largest a ÷ b: the biggest a, divided by the smallest b.
- For the smallest answer, swap them: the smallest a with the biggest b.
a = 30 and b = 6, to the nearest whole
Largest a ÷ b: 30.5 ÷ 5.5 = 5.55 (2 d.p.)
Smallest a ÷ b: 29.5 ÷ 6.5 = 4.54 (2 d.p.)
Watch out
Both upper bounds is a slip here: 30.5 ÷ 6.5 = 4.69, which is smaller than 30.5 ÷ 5.5.
Bounds & Rounding
HigherAnswers to a suitable accuracy
How do you use bounds to decide how accurately to give an answer?
Answer it in your head, then turn the card over.
Answers to a suitable accuracy
Key points
- Work out the upper and the lower bound of the answer.
- Round both to fewer and fewer figures until they agree.
- The true value lies between them, so it rounds to that value too.
- Quote both bounds and say that they agree: that is the justification.
Bounds 6.4183 and 6.4251
To 3 s.f.: 6.42 and 6.43, which differ
To 2 s.f.: 6.4 and 6.4, which agree
So the answer is 6.4 (2 s.f.)
Watch out
Giving more figures than the bounds agree on: if they differ at 3 s.f., stop at 2 s.f.
Print these cards
Four cards to an A4 sheet, each the size of a standard A6 revision card.
One-sided
Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
Put it into practice
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