# Common Sense In my many years of examining experience I have consistently observed a high proportion of students with implausible answers to questions, with no or insufficient thought as to the common sense of their answers.  This problem persists not only in the UK but globally, with little sign of improvement.

Students can always be relied upon to both amuse and frustrate me, usually in equal measures, with their improbable answers, for example in compound interest questions.  When investing £400 in a bank account at a rate of 3% per year compound interest for 5 years I would be delighted with a final investment value of £206 000 (from wrong working such as 400 × 103 × 5).  I would be bitterly disappointed with a final investment value less than £400, also frequently seen (e.g. from 400 × 0.035 = 9.72 × 10-6).

Many exam boards are changing question styles in exam papers, the types of questions where common sense is required to recognise the plausibility of answers are increasing.

We all conscientiously ask our students to ‘check their working’ but I am not convinced we devote enough time to giving them practice at this skill or advice on how precisely they can do this.  A very simple starting point, going back to the compound interest problem, is to ask students to remember to look back at the context of the problem and check that their answer is at least larger than the initial investment and then to check it is not ridiculously larger.  My most successful attempt to address this issue, with my own students, was to produce a compilation of common incorrect working leading to implausible answers. This is easily collected from tests from previous students, or it can be made up from your own experience.  This can be used frequently and reinforced in most number topics. It can be extended to other topics too.

Here is an exercise to give you an example starting point, which includes the cases above.

Hope it’s useful!

Debbie Barton

Cross out the two most unlikely answers without looking at the working.
Then with the two most plausible answers look at the working to check which is correct.

1 A school hall has 1540 seats.
Calculate the number of people in the hall when 55% of the seats are occupied.

Student A 693 0.45 × 1540
Student B 84 700 55 × 1540
Student C 4 (55 ÷ 1540) × 100
Student D 847 0.55 × 1540

Jamil invests £400 at a rate of 3% compound interest per year.
Work out the value of his final investment after 5 years.

Student A 206 000 400 × 103 × 5
Student B 0.00000972 400 × 0.035
Student C 463.71 400 × 1.035
Student D 2060 400 × 1.03 × 5

A hummingbird beats its wings 24 times per second.
Work out the number of times it beats its wings in one hour.

Student A 86 400 24 × 60 × 60
Student B 1.44 ×103 24 × 60
Student C 5.184 ×106 24 × 60 × 60 × 60
Student D 8.64 ×104 24 × 60 × 60

Find m when 3m ÷ 35 = √3

Student A 10 m ÷ 5 = 2
Student B 5.5 m – 5 = 0.5
Student C 2.5 m ÷ 5 = 0.5
Student D 4.5 m – 5 = 0.5

Sam has a rectangular drive measuring 4 m by 13 m.
He is covering his drive with gravel 7 cm deep.
Gravel is sold in 0.5m3 bags each costing £40.
Sam gets a discount of 20% off the cost of the gravel.
Work out how much Sam pays for the gravel.

Student A £23 296 4 × 13 × 7 = 364
364 ÷ 0.5 = 728
728 × 40 = 29 120
29 120 × 0.8
Student B £232.96 4 × 13 × 0.07 = 3.64
3.64 ÷ 0.5 = 7.28
7.28 × 40 = 291.2
291.2 × 0.8
Student C £256 4 × 13 × 0.07 = 3.64
3.64 ÷ 0.5 = 7.28 (=8)
8 × 40 = 320
320 × 0.8
Student D £32 400 × 1300 × 7 = 3 640 000
0.5 × 10003 = 500 000 000
3 640 000 ÷ 500 000 000 = 0.00728
1 × 40 = 40
40 × 0.8 Debbie Barton is a teacher, examiner and maths consultant with over 20 years’ experience. She’s written a number of books including Complete Mathematics for Cambridge Secondary 1. She also worked as a Gifted and Talented trainer and is passionate about ensuring able students are challenged with exciting mathematical stimulus.