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      Inequality notation to express error

      Lesson details

      Learning outcome

      I can calculate possible errors expressed using inequality notation 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.

      Keywords

      • Range - A range of values where every value in the range estimates to the given value.

      • 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

      Incorrect use of the error interval i.e $$a ≤ x ≤ b$$ or writing the error interval as $$a ≤ x > b$$

      As a check, encourage students to round the upper limit interval by the degree of accuracy to see if it evaluates to the rounded number. Also, draw a number line to show the meaning of $$a ≤ x > b$$ and allow students to see the error.

      Teacher tip

      The use of error intervals is used a lot in different industries around the world. Discuss why error intervals are so important. E.g If 5 m has been rounded to the nearest integer, why is it important to know that the actual measurement could have been 4.5 m? Why is it such a big deal?

      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.
      Which of the following is not an integer?

      −3249
      10 904
      Correct answer: 12 348.4839
      Correct answer: 34 839.48
      439 043 904

      Q2.
      State the smallest and largest integers that round to 520 when rounded to the nearest 10.

      Smallest integer = 514, largest integer = 525
      Smallest integer = 514, largest integer = 524
      Smallest integer = 515, largest integer = 525
      Correct answer: Smallest integer = 515, largest integer = 524

      Q3.
      State the smallest and largest integers that round to 3900 when rounded to the nearest 100.

      Smallest integer = 3849, largest integer = 3949
      Correct answer: Smallest integer = 3850, largest integer = 3949
      Smallest integer = 3850, largest integer = 3950
      Smallest integer = 3855, largest integer = 3950

      Q4.
      When does 0.4999999 NOT round to give 0.5?

      Correct answer: When rounded to the nearest integer.
      When rounded to 1 significant figure.
      When rounded to 1 decimal place.

      Q5.
      Work out 3.4 km + 348 m + 6000 cm. Give your answer in metres.

      Correct answer: 3808 m
      380.8 m
      38.08 m
      4 348 m
      434.8 m

      Q6.
      Work out 0.05 km + 6.32 m + 123 cm. Give your answer in centimetres.

      80.5 cm
      805 cm
      1255 cm
      Correct answer: 5755 cm
      12 550 cm

      6 Questions

      Q1.
      An error interval for a number $$x$$ shows the range of possible values of $$x$$. It is written as:

       $$ a< x < b$$
      $$ a> x < b$$
      $$a \geq x <b$$
      Correct answer: $$ a\leq x < b$$

      Q2.
      When a number is rounded to the nearest integer the result is 8. Identify the correct range for the unrounded number.

      Correct answer: Smallest = $$7.5$$, Largest = $$8.4 \dot 9$$
      Smallest = $$7.5$$, Largest = $$8.9 \dot 9$$
      Smallest = $$7.5$$, Largest = $$8.5 \dot 9$$
      Smallest = $$7.5$$, Largest = $$8.5$$

      Q3.
      When a number is rounded to the nearest integer the result is 19. Identify the correct range for the unrounded number.

      Smallest = $$18.5$$, Largest = $$19.5 \dot 9$$
      Correct answer: Smallest = $$18.5$$, Largest = $$19.4 \dot 9$$
      Smallest = $$17.5$$, Largest = $$19.5$$
      Smallest = $$18.5$$, Largest = $$19.5$$

      Q4.
      When a number, $$x$$, is rounded to the nearest integer the result is 8. Identify the error interval for $$x$$.

       $$ 7.5< x < 8.4 \dot 9$$
      $$ 7.5> x < 8.4 \dot 9$$
      $$ 7.5\leq x < 8.4 \dot 9$$
      Correct answer: $$7.5\leq x< 8.5$$
      $$7.5 \geq x <8.4 \dot 9$$

      Q5.
      When a number, $$x$$, is rounded to 2 significant figures the result is 7.8. Identify the error interval for $$x$$.

       $$ 7.75< x < 7.84 \dot 9$$
      Correct answer:  $$ 7.75\leq x < 7.85 $$
       $$ 7.74 \dot 9< x < 7.84 \dot 9$$
       $$ 7.75 < x < 7.85 $$

      Q6.
      A distance $$x$$ km is given as 2.345 km correct to the nearest metre. Identify the error interval for $$x$$.

      $$ 2.3445\leq x < 2.3449$$
      $$2.3445 \geq x <2.3454 \dot 9$$
      $$ 2.3445> x < 2.3454\dot 9$$
       $$ 2.3445< x < 2.3455$$
      Correct answer: $$ 2.3445\leq x < 2.3455$$

      To help you plan your 8 maths lesson on: Inequality notation to express error, download all teaching resources for free and adapt to suit your pupils' needs...