EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Accuracy
More trigonometry · Lesson 1 of 9
Warm-up
Answer each one, then check.
1. Round 3.47 to 1 decimal place.
3.5
2. Round 4650 to the nearest 100.
4700
3. What does < mean?
Less than
4. What does ≤ mean?
Less than or equal to
5. Round 0.0364 to 2 significant figures.
0.036
Learning Objectives
1. Find the upper and lower bounds of a rounded number.
2. Write an error interval using inequalities.
3. Find bounds for sums, differences, products and quotients (Higher).
4. Choose a sensible degree of accuracy for an answer.
Bounds
A rounded value could have started anywhere in a range: half a unit either side of it.
If a number is rounded to the nearest 10, the range is 5 either side. Nearest 1: 0.5 either side. Nearest 0.1: 0.05 either side.
Bounds on a Number Line
|
Closed circle: included. Open circle: not included. |
A number line showing that 45 to the nearest 5 could be anywhere from 42.5 up to but not including 47.5.
Finding Bounds
|
A length is 8.3 cm, rounded to 1 decimal place. Write down the lower bound and the upper bound. |
1. Rounding unit
0.1, so half a unit is 0.05
2. Lower bound
8.3 − 0.05 = 8.25
3. Upper bound
8.3 + 0.05 = 8.35
Answer: Lower bound 8.25 cm, upper bound 8.35 cm.
Writing an Error Interval
|
n = 350 to the nearest 10. Write the error interval for n. |
1. Half a unit
5
2. Lower bound is included
345 ≤ n
3. Upper bound is not included
n < 355
Answer: 345 ≤ n < 355
Rounding and Bounds
Half a unit each way.
|
Rounded to |
Value |
Error interval |
|---|---|---|
|
Nearest 10 |
60 |
55 ≤ x < 65 |
|
Nearest whole number |
7 |
6.5 ≤ x < 7.5 |
|
1 decimal place |
2.4 |
2.35 ≤ x < 2.45 |
|
2 significant figures |
0.83 |
0.825 ≤ x < 0.835 |
Bounds in Calculations (Higher)
|
BIGGEST POSSIBLE ANSWER |
SMALLEST POSSIBLE ANSWER |
|
▸ Add: use both upper bounds. ▸ Subtract: upper bound minus lower bound. ▸ Multiply: upper times upper. ▸ Divide: upper bound divided by lower bound. |
▸ Add: use both lower bounds. ▸ Subtract: lower bound minus upper bound. ▸ Multiply: lower times lower. ▸ Divide: lower bound divided by upper bound. |
Bounds of a Rectangle's Area (Higher)
|
A rectangle has length 12 cm and width 5 cm, both to the nearest centimetre. Find the greatest possible area. |
1. Upper bound of length
12.5
2. Upper bound of width
5.5
3. Multiply
12.5 × 5.5 = 68.75
Answer: 68.75 cm²
Sensible Accuracy
Do not give an answer more accurately than the data allows.
▸ Round at the end. Keep full calculator values until the last step.
▸ Match the data. If measurements are to 2 significant figures, do not quote 6 in the answer.
▸ Bounds agree. If the upper and lower bounds of an answer round to the same value, that value is the answer to that accuracy.
Key Terms
|
Lower bound The smallest value that rounds to the given number. |
Upper bound The value at the top of the range; not itself included. |
|
Error interval The range of possible values written with inequalities. |
Degree of accuracy How precisely a value is given, e.g. 1 d.p. or 2 s.f. |
|
Truncate Cut off digits without rounding. |
Significant figures Digits that carry meaning, counting from the first non-zero digit. |
Your Task: Bounds Race
10 minutes
|
Find the error interval for each value. (a) 6.8 to 1 d.p. (b) 1200 to the nearest 100 (c) 0.05 to 1 significant figure (d) 25 to 2 significant figures. 1. Find the rounding unit. 2. Halve it and add and subtract. |
A good answer shows: (a) 6.75 ≤ x < 6.85 (b) 1150 ≤ x < 1250 (c) 0.045 ≤ x < 0.055 (d) 24.5 ≤ x < 25.5
Can I...?
☐ Find the rounding unit.
☐ Write a lower bound.
☐ Write an upper bound.
☐ Write an error interval.
☐ Use the right inequality signs.
☐ Find bounds for an addition or subtraction.
☐ Find bounds for a product or quotient.
☐ Choose a sensible accuracy.
Summary
✓ Half a unit each side of the rounded value.
✓ Lower bound included, upper bound not.
✓ Bounds calculations: choose the extreme values that make the answer largest or smallest.
✓ Do not round until the final answer.
|
EXAM FOCUS x = 7.4 to 1 decimal place. Write down the error interval for x. (2 marks) Lower bound uses ≤; upper bound uses <. |