## 13 April 2018

### New GCSE: Bounds

I like the bounds content in the new GCSE. I think that the introduction of error intervals has provided some clarity. Previously some students just couldn't get their head around why we use 3.65 as the upper bound for 3.6 ("but Miss, 3.65 rounds to 3.7. This doesn't make sense!"). The use of inequality symbols (in conjunction with instruction using number lines) really helps with conceptual understanding here.
3.55 ≤  x < 3.65

Of course error intervals aren't new - they just weren't previously assessed at GCSE. Upper bounds and error intervals are clearly explained in this extract from the CIMT MEP textbook chapter on Estimation and Approximation from the late 1990s:
Truncation
I also like the inclusion of truncation in the new GCSE specification, because it means that students need to think more carefully before answering a bounds question. Previously, bounds questions at GCSE were predictable, and students only really needed a superficial, procedural understanding of the topic. Things have changed.

Age is a good way to teach truncation, given that students will already be very used to truncating their own age, rather than rounding it to the nearest integer.
Ed is 36 years old. His age is represented by x. Give the error interval for x.
36 ≤  x < 37

Discrete Bounds
My Year 11s really struggled with a bounds question in their mock exams this year. It's from a recent AQA paper so I can't share the actual question here, but this question is similar:
Two integers are rounded to the nearest 10. The rounded numbers are added to give to 40. What's the maximum total value of the orginal numbers?
Let's say that both numbers were 20 when rounded. The maximum each number could be is 24, so the highest possible total is 48. The common mistake here is to take the upper bound to be 25, giving an incorrect final answer of 50. So why was this such a common mistake? I expect it's because most of us teach bounds in a context of measurement, not counting. We spend a lot of time on continuous bounds, and very little (if any) time on discrete bounds.

Interestingly, my GCSE textbook doesn't contain a single question on discrete bounds. A bit of internet searching shows that BBC Bitesize has one question on discrete bounds:
The number of people on a bus is given as 50, correct to the nearest 10. What is the lowest and highest possible number of people on the bus?
The OCR Check-In Test on Approximation and Estimation has this question:
Explain why the error interval of 400 cars to the nearest 50 cars could be written as  375 ≤  c  424 or 375 ≤  c < 425.
and the AQA Rounding Topic Test has a couple of discrete bounds questions, including this one:
Two performances of a show are each attended by 175 people, to the nearest 5. Work out the maximum possible difference between the numbers of people attending.

It's worth noting that the Government's GCSE subject content doesn't specifically refer to discrete bounds. This is the official content for bounds at GCSE:

"15. round numbers and measures to an appropriate degree of accuracy (e.g. to a specified number of decimal places or significant figures); use inequality notation to specify simple error intervals due to truncation or rounding

16. apply and interpret limits of accuracy, including upper and lower bounds"

OCR's specification does mention discrete bounds though, saying that students must "Understand the difference between bounds of discrete and continuous quantities".

AQA's excellent Teaching Guidance has this:
"Upper bounds do not necessarily require use of recurring decimals. For example, if the answer to the nearest integer is 7, the maximum could be given as 7.5, 7.49.... , or 7.49.
If this value of 7 represented £7, £7.49 would be expected for the maximum.
For continuous variables, students may be asked for the lower and upper limits rather than the minimum and maximum values."
The money example here is interesting - it's not something I have explicitly covered with my students. Don Steward has a good bounds exercise including money questions - I'll  make sure I use this next time I teach this topic.

Edexcel doesn't specifically refer to discrete bounds in either their specification or supporting material, but of course that doesn't necessarily mean they won't come up in an Edexcel exam.

Exam Questions
Bounds questions involving calculations can be fairly challenging for students, particularly as sometimes it's not immediately obvious that the question involves bounds. Here's an example from an old Edexcel Linked Pair paper:
"Sian is driving on a motorway.
Sian drives for 2.8 miles, correct to the nearest tenth of a mile.
It takes her 200 seconds, correct to the nearest 5 seconds.
The average speed limit on this part of the motorway is 50 miles per hour.
Did Sian drive at a speed within the average speed limit? You must explain your answer."

Another type of challenging GCSE question is one that says "by considering bounds, work out the value of x to a suitable degree of accuracy, justifying your answer". For examples of questions like this, see Dr Frost's 'Full Coverage: Bounds' resource which has an example of every different question type from past Edexcel papers.

Resources
I've already mentioned Don's excellent resource, the CIMT resources and Dr Frost's full coverage GCSE questions. For a comprehensive list of bounds resources, see my resources library.

It's worth noting that John Corbett has helpful videos, textbook exercises and practice questions on limits of accuracy and limits of accuracy: applying. And Edexcel's new content resources include a helpful worksheet on bounds.

I'm doing a revision lesson on bounds with my Year 11s on Monday and will be using this excellent bounds GCSE revision resource from Maths4Everyone.

For Interest
Finally, in my research into how mathematics was taught in the 18th, 19th and 20th centuries, I've found few references to rounding. However, in 'Practical Mathematics for All' (McKay, 1942) I found this nice explanation of 'limits of error':

If anyone knows of any earlier references to upper and lower bounds, I'd love to see them.

Thanks for reading! This was my 300th blog post, so a bit of a milestone for me. To read other posts about teaching specific GCSE topics, you can view my topic collection here.

1. I am confused by the AQA guidance you quoted:

"Upper bounds do not necessarily require use of recurring decimals. For example, if the answer to the nearest integer is 7, the maximum could be given as 7.5, 7.49.... , or 7.49.
If this value of 7 represented £7, £7.49 would be expected for the maximum..."

I understood the convention to be that the range of values is strictly less than the upper bound (x < UB). In the example though, you would not have x < £7.49?

Also can you give an example where recurring decimals are used as a bound?

1. I think AQA's point is that if a continuous value is given as '7 to the nearest integer' then we could give 7.5 as the upper bound, or we could give 7.49recurring which is the same thing as 7.5 (that's a fun thing to prove to students). I would never use the recurring decimal bound, but it's no different to using 7.5 so it's ok to do so.

Regarding the £7.49: I think their point is not about error intervals (which we would only use for continuous data) but about the value you would use in a calculation requiring an upper bound.

2. On looking more carefully, I realize I missed the mark (dot/dash) indicating a recurring '9' above the final 9 in the list "7.5, 7.49.... , or 7.49" as shown in the original post. When I googled that specific line of text, I can see that others have omitted the dash/dot as well when copying this text.

Therefore I read that final ", or 7.49" as an alternative upper bound, not another way of writing "7.49...". This implied that 7.49 was also a possible upper bound, which is not correct. I hope that makes sense!

Separately I am unclear as to why they have used the term "maximum" rather than "upper bound" in the text...

1. Oh I see! The AQA guidance, as in my post, has the following three options:
7.5
7.49... (with the dots)
7.49recurring (I used a vinculum because I couldn't work out how to do the dot in html).

Not sure about the word maximum.

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