Rounding, bounds and standard form
Real measurements are never exact. Bounds tell you how far a rounded value could be from the truth, and standard form keeps very large and very small numbers manageable.
Lesson overview
What you'll learn in this lesson
Apply number operations, accuracy and standard form to solve GCSE-style problems.
Key learning points
- • Significant figures and estimation
- • Upper and lower bounds
- • Standard form
This lesson at a glance
- 30 minutes
- 16 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 16
Watch first
Rounding, bounds and standard form
An English teaching video on this topic from Cognito. Watch it, then scroll down for the lesson.
In a nutshell
Apply number operations, accuracy and standard form to solve GCSE-style problems.
Learning cycle
Part 2 of 16
Learning cycle 1 of 2
Part 1 · Significant figures and estimation
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 3 of 16
Learn
Significant figures and estimation
The first significant figure is the first non-zero digit. To estimate a calculation, round each value to one significant figure and work it out mentally: 38.7 × 6.2 ÷ 0.48 becomes 40 × 6 ÷ 0.5 = 480. Examiners award the method, so show the rounded values.
Explore the idea
Part 4 of 16
Learn
Upper and lower bounds
A length given as 24 cm to the nearest centimetre lies between 23.5 cm and 24.5 cm. For a sum or product, the maximum uses both upper bounds. For a subtraction or division, the maximum uses the upper bound of the first and the lower bound of the second — a rule worth memorising.
Quick check
Part 5 of 16
Quick check
0.00482 to 2 significant figures is
Quick check
Part 6 of 16
Quick check
A mass of 60 kg to the nearest kg has lower bound
Reset break
Part 7 of 16
Pause
That's sitting 1 of 2 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Learning cycle
Part 8 of 16
Learning cycle 2 of 2
Part 2 · Standard form
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 9 of 16
Learn
Standard form
A number in standard form is written as A × 10ⁿ, where 1 ≤ A < 10. Multiply by adding the indices and dividing by subtracting them, then adjust so A sits back in range: 40 × 10⁵ must be rewritten as 4 × 10⁶ before it counts as standard form.
Quick check
Part 10 of 16
Quick check
(3 × 10⁵) × (4 × 10³) equals
Quick check
Part 11 of 16
Quick check
For a division, the upper bound uses
Explore the idea
Part 12 of 16
Worked example
Worked answer: a rectangle measures 8.4 cm by 5.2 cm, each to 1 d.p. Find the upper bound of its area (3 marks)
The upper bounds of the sides are 8.45 cm and 5.25 cm, since each measurement is within 0.05 cm of the stated value (1). Area is a product, so the upper bound uses both upper bounds: 8.45 × 5.25 (1) = 44.3625 cm² (1). Using 8.5 and 5.3 is the common error: those are bounds to the nearest 0.1, not the nearest 0.05.
Challenge round
Part 13 of 16
Game · Sort it
Which of these are true?
Drag each card into the right column. Tap a card first if dragging is fiddly.
True
Not true
Challenge round
Part 14 of 16
Game · Fill the gaps
Finish the sentences
Choose the word that belongs in each gap.
A ____ in standard form is written as A × 10ⁿ, where 1 ≤ A < 10.
Multiply by adding the indices and dividing by subtracting them, then adjust so A sits back in range: 40 × 10⁵ must be rewritten as 4 × 10⁶ before it counts as ____ form.
Mastery quiz
Part 15 of 16
Marked quiz
End of lesson quiz: Rounding, bounds and standard form
4 questions, marked with the reasoning shown. No timer.
1. 0.00482 to 2 significant figures is
2. A mass of 60 kg to the nearest kg has lower bound
3. (3 × 10⁵) × (4 × 10³) equals
4. For a division, the upper bound uses
Lesson round-up
Part 16 of 16
Lesson round-up
Ready when you are
Quiz score
Not sat
Games
Not played
Points this lesson
0
Best run
0 in a row
Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?
Ask LunaDo the work
Your written work for this lesson
This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.
The cost of a school trip
A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.
If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.
The school decides to charge £20 and to use anything left over for a second trip later in the year.
Written for Good Learning Co.
Reading questions
- Work out the total cost of the trip if 50 pupils go.
- Show why the price per pupil falls when more pupils go, using numbers.
- At £20 each, how many pupils must go before the trip makes money? Show your working.
Working out
Working outDo the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.
10–15 questions
Worked example, explained
WritingCopy one question out and write underneath, in words, why each step works.
1 question, 4–5 sentences
Problem in words
Working outWrite your own real-life problem that uses today's method, then solve it.
1 problem + solution
Saved as you type.
Part 1 of 16 · Watch & discover