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      Lesson details

      Learning outcome

      I can calculate the lower and upper bounds of values and express these as a ≤ x < b.

      Key learning points

      1. When given a rounded number you can say what the biggest and smallest values of the unrounded number could have been .
      2. You can use a ≤ x < b notation to express the range of values the unrounded number could have taken.
      3. You can use those values in your estimate to see what your biggest and smallest answers could have been.
      4. The range of possible answers is the error interval.

      Keywords

      • Upper bound - The upper bound for a rounded number is the smallest value that would round up to the next rounded value.

      • Lower bound - The lower bound for a rounded number is the smallest value that the number could have taken prior to being rounded.

      • Error interval - An error interval for a number x shows the range of possible values of x. It is written as an inequality a ≤ x < b

      Common misconception

      For example if a measurement of 12 cm has been given to the nearest centimetre assuming the upper bound is 12.4 cm.

      Emphasise the infinite nature of a measurement.

      Teacher tip

      Use MWBs, give pupils an error interval and ask them to write down a number which lies in the error interval. Keep making the error interval smaller this will allow pupils to see that there is always a number no matter how small the interval is and help with the misconception.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      In the number 56 392 the 3 is the third figure.

      Correct Answer: significant

      Q2.
      Choose the correct calculation to estimate the cost of hiring an electrician for 4.5 hours, where the total cost = £42.50 × number of hours + £25

      £50 × 5 + £30
      £50 × 5 + £20
      £50 × 4 + £30
      Correct answer: £40 × 5 + £30
      £40 × 4 + £30

      Q3.
      Estimate the value of $${435.8\times8.742\over3.04}$$

      Correct Answer: 1200

      Q4.
      Will the estimation of 128.03 + 0.512 be an overestimate or underestimate?

      Correct Answer: overestimate, over estimate

      Q5.
      Estimate the amount of change received from £500 when purchasing a TV costing £375

      Correct Answer: £100, 100

      Q6.
      Estimate the circumference of a circle with a radius of 8.4 cm.

      Correct Answer: 60 cm, 60, 60cm

      6 Questions

      Q1.
      Upper and lower bounds are used when a number has been .

      Correct Answer: rounded

      Q2.
      Work out the lower and upper bounds of 320 rounded to the nearest 10

      319.5 and 320.5
      315 and 324
      310 and 330
      Correct answer: 315 and 325
      310 and 325

      Q3.
      Work out the lower and upper bounds of 12.6 rounded to 1 decimal place.

      12.7 and 12.9
      12.1 and 13.1
      Correct answer: 12.55 and 12.65
      12.55 and 12.64
      12.55 and 12.6499999999999

      Q4.
      rounded to the nearest integer. The lower bound is 147.5 and the upper bound is 148.5

      Correct Answer: 148, 148.0

      Q5.
      rounded to 1 decimal place. The lower bound is 10.25 and the upper bound is 10.35

      Correct Answer: 10.3, 10.30

      Q6.
      rounded to 2 significant figures. The lower bound is 8750 and the upper bound is 8850

      Correct Answer: 8800, 8 800

      To help you plan your 10 maths lesson on: Upper and lower bounds, download all teaching resources for free and adapt to suit your pupils' needs...