EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Units and accuracy
Area and volume · Lesson 2 of 7
Last Lesson and Before
Answer each one, then check.
1. How many cm are in 1 m?
100
2. How many mm are in 1 cm?
10
3. Round 8.347 to 1 decimal place.
8.3
4. Last lesson: area of a 3 m by 2 m rectangle?
6 m²
Learning Objectives
1. Convert between units of area.
2. Convert between units of volume and capacity.
3. Write the error interval for a rounded or truncated number.
4. (Higher) Find upper and lower bounds of a calculation.
Why 1 m² Is 10 000 cm²
|
A square metre is 100 cm by 100 cm, so it holds 100 × 100 = 10 000 square centimetres. A cubic metre is 100 × 100 × 100 = 1 000 000 cubic centimetres. |
Square the length factor for area; cube it for volume. |
Conversion Factors
|
Units |
Length |
Area |
Volume or capacity |
|---|---|---|---|
|
mm and cm |
1 cm = 10 mm |
1 cm² = 100 mm² |
1 cm³ = 1000 mm³ |
|
cm and m |
1 m = 100 cm |
1 m² = 10 000 cm² |
1 m³ = 1 000 000 cm³ |
|
m and km |
1 km = 1000 m |
1 km² = 1 000 000 m² |
Rarely needed |
|
Capacity |
- |
- |
1 cm³ = 1 ml; 1000 cm³ = 1 litre; 1 m³ = 1000 litres |
Converting Area
|
Convert 3.5 m² into cm². |
1. 1 m² = 10 000 cm²
Multiply by 10 000 (a smaller unit means more of them)
2. Calculate
3.5 × 10 000 = 35 000
Answer: 35 000 cm²
Converting Volume to Capacity
|
A tank holds 2 400 000 cm³ of water. Write this in m³ and in litres. |
1. 1 m³ = 1 000 000 cm³
2 400 000 ÷ 1 000 000 = 2.4 m³
2. 1 litre = 1000 cm³
2 400 000 ÷ 1000 = 2400 litres
3. Check: 1 m³ = 1000 litres
2.4 × 1000 = 2400
Answer: 2.4 m³, which is 2400 litres
PART TWO
Accuracy
A rounded measurement could be anywhere in a range.
Error Intervals
If a length is 8.3 cm to 1 decimal place, it could be anything that rounds to 8.3.
▸ The bounds. Half a unit either side: the lower bound is 8.25 and the upper bound is 8.35.
▸ Error interval. 8.25 ≤ l < 8.35. The upper bound itself would round up to 8.4, so use <.
▸ To the nearest 10. 70 to the nearest 10: 65 ≤ x < 75.
▸ Truncation. 4.7 truncated to 1 d.p. (digits chopped off): 4.7 ≤ x < 4.8.
An Error Interval
|
The mass of a parcel is 2.6 kg, correct to 1 decimal place. Write down the error interval for the mass m. |
1. The degree of accuracy
0.1 kg
2. Half of it
0.05 kg
3. Lower and upper bounds
2.6 − 0.05 = 2.55 and 2.6 + 0.05 = 2.65
Answer: 2.55 ≤ m < 2.65
PART THREE · HIGHER
Bounds in Calculations
Choose the bounds that push the answer as far as it can go.
Which Bounds to Use
|
Calculation |
For the upper bound |
For the lower bound |
|---|---|---|
|
a + b |
upper + upper |
lower + lower |
|
a − b |
upper − lower |
lower − upper |
|
a × b |
upper × upper |
lower × lower |
|
a ÷ b |
upper ÷ lower |
lower ÷ upper |
The Upper Bound of a Speed
|
A runner covers 100 m, measured to the nearest metre, in 12.5 s, measured to 1 decimal place. Work out the upper bound of the runner's average speed. |
1. Bounds of the distance
99.5 m and 100.5 m
2. Bounds of the time
12.45 s and 12.55 s
3. Biggest speed: biggest distance ÷ smallest time
100.5/12.45 = 8.0722…
Answer: 8.07 m/s (to 3 s.f.)
Key Terms
|
Capacity How much a container holds, in ml or litres. |
Lower bound The smallest value that rounds to the given number. |
|
Upper bound The value where rounding switches up to the next number; the interval stops just below it. |
Error interval The range of possible values, e.g. 8.25 ≤ l < 8.35. |
|
Truncate Cut off digits without rounding. |
|
Your Task: Fill the Room
12 minutes
|
Measure (or estimate) your classroom's length, width and height in metres. Work out its volume in m³ and in litres. Then work out its floor area in cm². Which of your answers are sensible to give exactly, and which should be rounded? 1. Measure in metres. 2. Convert with the right factor. 3. Decide how accurate your answer really is. |
A good answer shows: For a 9 m by 7 m by 3 m room: volume 189 m³ = 189 000 litres; floor area 63 m² = 630 000 cm². With measurements to the nearest metre, the answers are only good to 2 significant figures at most.
Can I...?
☐ Convert between units of area.
☐ Convert between units of volume.
☐ Convert between volume and capacity.
☐ Write an error interval for a rounded number.
☐ Write an error interval for a truncated number.
☐ (Higher) Find the bounds of a calculation.
Summary
✓ Area: square the length factor. Volume: cube it.
✓ 1 cm³ = 1 ml; 1000 cm³ = 1 litre; 1 m³ = 1000 litres.
✓ Error interval: half a unit either side, ≤ then <.
✓ (Higher) Maximise or minimise each part of the calculation.
|
EXAM FOCUS The length of a pencil is 14 cm, correct to the nearest centimetre. Write down the error interval for the length l. (2 marks) In an error interval the lower bound gets ≤ and the upper bound gets <. Writing ≤ at both ends loses a mark. |