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Maths

Upper and Lower Bounds

Overview

Students develop a rigorous understanding of measurement accuracy, applying bounds to compound measures to assess the maximum possible error in calculated results.

Learning Objective
Students find upper and lower bounds for rounded measurements and use them to determine the bounds for calculated quantities, expressing answers to an appropriate degree of accuracy.

Resources needed

  • Calculator
  • Mini whiteboards

Lesson stages

0 / 7 done
  1. 1 A length is given as 7.4cm (to 1 d.p.). 'What could the actual length be?' Students discuss. Establish: anywhere from 7.35cm (would round up to 7.4) to just below 7.45cm (would round down to 7.4). Introduce: lower bound = 7.35, upper bound = 7.45.
  2. 2 Rule: half the degree of accuracy up and down from the rounded value. 7.4cm (to 1 d.p., accuracy = 0.1): bounds are 7.35 and 7.45. Practice: find bounds for 5.2m, 180g, 3.00kg (to 2 d.p.), 400cm (to nearest 10).
  3. 3 Two lengths: A = 8.4cm, B = 5.7cm. Min total = 8.35+5.65 = 14.00cm. Max total = 8.45+5.75 = 14.20cm. For the maximum sum, use upper bounds for both.
  4. 4 Speed = distance ÷ time. Distance = 48km (nearest km), Time = 1.2h (1 d.p.). Maximum speed: use maximum distance ÷ minimum time. Min speed: use minimum distance ÷ maximum time. Students must decide which bound to use for numerator and denominator.
  5. 5 Which bounds to combine depends on what you want to maximise or minimise. For maximum (sum, product): use upper bounds. For minimum (sum, product): use lower bounds. For max quotient: upper ÷ lower. For min quotient: lower ÷ upper.
  6. 6 Area of a rectangle: l = 9.6cm, w = 4.3cm. Find max and min area. Also: a percentage — if 34 out of 97 (both to nearest whole): find max and min percentage.
  7. 7 Why does this matter in real life? (Engineering tolerances, medical dosages, scientific measurements.) What happens if you always use the nominal values without bounds?

Tap a step to mark it as done.

Variations

  • Extend to bounds in trigonometry (angles rounded)
  • Error intervals with inequality notation
  • Absolute and relative error
More information

Display: bound, upper bound, lower bound, error interval, degree of accuracy, tolerance. Notation: 7.35 ≤ x < 7.45.

Focus on addition/subtraction bounds before multiplication/division. Provide a decision flowchart: 'To maximise, use upper/upper for × and ÷ upper by lower.'

Do students apply ±½ of the accuracy unit correctly? Do they identify the correct bound combination for each operation? Do they express the error interval with correct inequality notation?

All calculations by hand or basic calculator. No printed sheets needed — problems presented on board.

Students may use 7.35 and 7.4 (the original value) as bounds rather than 7.35 and 7.45. Stress: the upper bound is half the accuracy unit above the rounded value.

Activity details
Secondary Challenging 45 minutes 5 minutes prep Solo, Pairs Indoor
Topics
Bounds Error Rounding Measurement Estimation