Study case: finding an error interval for a rounded measurement In this GCSE Bounds Error Interval Problem Tutor, I use one study case to show how bounds become logical when the student thinks about rounding. The problem is: a length is measured as 7.4 cm to the nearest tenth of a centimetre. Write the error interval for the true length. The answer is 7.35 <= x < 7.45. I like this example because it is simple enough to understand, but it teaches the exact boundary language that many GCSE students find difficult. Before solving, I ask the student what nearest tenth means. A tenth of a centimetre is 0.1 cm. Half of 0.1 is 0.05, so the true value can be 0.05 below or 0.05 above 7.4 before it would round to a different tenth. This is the most important idea in bounds: use half of the rounding unit. Once the student understands that, the interval becomes much easier. Method 1 is the half-step method. Start with 7.4. Subtract 0.05 to find the lower bound: 7.35. Add 0.05 to find the upper boundary: 7.45. The lower bound is included because 7.35 rounds to 7.4 to the nearest tenth. The upper boundary is not included because 7.45 would round to 7.5, not 7.4. Therefore the error interval is 7.35 <= x < 7.45. I make the student say why the inequality signs are different. Method 2 is the number line method. I draw a number line around 7.4 and mark 7.35, 7.4 and 7.45. Values from 7.35 up to but not including 7.45 round to 7.4. This visual method is useful because it shows the interval as a range, not two separate numbers. It also helps students understand why the upper boundary is open. Method 3 is checking by rounding examples. I choose a value inside the interval, such as 7.42. Rounded to the nearest tenth, it becomes 7.4. I then choose a value just below the interval, such as 7.34, which rounds to 7.3. I choose the upper boundary 7.45, which rounds to 7.5. This check confirms that the interval has been written correctly. I use this method when a student is unsure about inclusive and exclusive boundaries. After the methods, I compare them. The half-step method is the fastest exam method. The number line method gives the clearest understanding. Checking examples protects against incorrect inequality signs. I want students to know all three because bounds questions often become harder when the rounded values are used in area, speed, density or percentage calculations. The most common mistakes are using the whole rounding unit instead of half, writing 7.35 < x < 7.45 and excluding the lower bound, including the upper bound, or confusing decimal places with significant figures. I use these mistakes to diagnose the issue. If the rounding unit is wrong, we practise identifying the accuracy first. If the inequality signs are wrong, we return to rounding examples. This topic is important because GCSE bounds questions often reward clear interval notation. A student may understand the idea but lose marks if the notation is careless. I encourage students to write the variable first, for example let x be the true length, and then write the full interval. This makes the answer mathematically complete. A useful extension is to find the error interval for 240 rounded to the nearest 10, which would be 235 <= x < 245. Another extension is to use two bounded measurements to find the greatest or least possible area. These extensions show why bounds matter in real measurement situations. In one-to-one tutoring, I would finish by giving the student three rounded values with different accuracies: nearest tenth, nearest whole number and nearest 5. I would ask them to identify the rounding unit first, then find half of it, then write the interval. This structured routine builds confidence and reduces careless errors. For support with this topic, students can book one-to-one tutor services: https://www.mastermathstutoring.co.uk/services. For the wider course page, visit GCSE and IGCSE Maths tutoring: https://www.mastermathstutoring.co.uk/gcse-igcse. To ask about lessons or availability, use contact MasterMaths Tutoring: https://www.mastermathstutoring.co.uk/contact-me.