Functional Maths
Analysing solutions
These activities and page references are from Teaching and Learning Functional Maths written by Susan Wall to support the pilot for functional mathematics.
Asking learners to mark their own work or that of another learner is a powerful way of encouraging them to think beyond the answer to a problem and to become reflective and selfcritical. This can be done on any piece of work that learners complete, from a practice exercise to a complex problem. When assessing, learners should be invited to write advice to the person who has tackled the problem. This puts the learner in a critical, advisory role. In activities of this kind, learners will be interpreting solutions.
Once learners are used to selfassessing and peer assessing, they can be given more substantial problems to assess such as the ones on pages 61–67. These problems contain some good ideas, some poor ideas and some superfluous ideas. Learners should be encouraged to identify the three types. It may be that not all learners will come up with the same opinions but, so long as they can justify them, all opinions are accepted. This again reinforces the idea that not all solutions to complex problems will necessarily follow the same route.
When asking learners to assess madeup solutions such as these, it is a good opportunity to introduce common misconceptions and errors. For example, in problem 2 (page 64), the solution takes the route that many learners do when doing a statistical analysis, i.e. starting by drawing bar and pie charts and calculating all averages whether or not they are appropriate.
Problem 1
Design a box for 30 chocolates. Each chocolate is cylindrical with diameter 1.5 cm and height 1 cm. Without including any flaps, how much card will the design need?
Possible solution
I am going to design a squarebased pyramid shape.
Volume of each chocolate 
= πr^{2}h = π × 0.75 × 0.75 × 1 

= 1.34 cm^{3} 


The volume of space needed for each chocolate 
= 1.5 × 1.5 × 1 cm^{3} 

= 2.25 cm^{3} 
Arrange them like:
Bottom layer 
Second layer 


Third layer 
Fourth layer 
Top layer 



From the top it will look like:
From the side it will look like:
Choose dimensions of triangle to be:
base: 8cm
height: 7cm
Using Pythagoras: 8^{2} + 3.5^{2} = 76.25
√76.25 = 8.73 cm
This is the net of my final design:
Area of triangle = ½ x 8 x 7 = 28cm^{2}
Area of square = 8 x 8 = 64cm^{2}
Area of card needed = 92cm^{2}
Ask learners to look through this solution to the problem.
Ask them to comment on it and give opinions on these issues:
 Which calculations are appropriate and which are not appropriate, and why?
 Are the diagrams helpful?
 Does the design work?
 Are the decisions clearly explained?
 Could you suggest ways in which to improve how the solution has been presented?
 Could you improve the design?
Problem 2
Use the information below to investigate women’s earnings in relation to men’s earnings.
Women’s earnings as a percentage of men’s in Great Britain
Year 
Percentage 
Year 
Percentage 
1970 
54 
1985 
68 
1971 
56 
1986 
66 
1972 
56 
1987 
66 
1973 
55 
1988 
67 
1974 
56 
1989 
68 
1975 
62 
1990 
68 
1976 
64 
1991 
70 
1977 
65 
1992 
71 
1978 
63 
1993 
71 
1979 
62 
1994 
72 
1980 
63 
1995 
72 
1981 
65 
1996 
72 
1982 
64 
1997 
73 
1983 
66 
1998 
72 
1984 
66 
1999 
74 
Source: ONS Social Trends
Possible solution
I am going to draw a bar chart of the percentages:
66% was the most common percentage.
A lot of percentages only happened once.
I am going to group the percentages and draw a pie chart:
There were more percentages in the 60s than in the 50s or 70s.
The 70s had the second most.
I am going to calculate the mean, mode and median of the percentages.
Mean = 1965 ÷ 30 = 65.5%
Mode = 66%
Median = 66%
The mean percentage over the period was 65.5%.
There were more 66% than any other percentage.
I am going to plot a line graph to show the trend over time:
Women's as percentages of men's
There was a relatively huge rise between 1974 and 1977.
I think this graph is a bit misleading because it implies that there were a lot of big changes whereas the changes were only in single figures.
I am going to redraw it with different axes:
Women's as percentage of men's
The trend is upwards.
Overall the percentage has risen by 20% over 20 years. The percentage has risen at an average rate of 1% per year. Average wages have risen a lot in this period of time therefore there is still a big gap between men’s and women’s wages.
More women are career women now and there are more women in top jobs. Therefore there must be still a lot of women at the bottom end who are very poorly paid.
Maternity leave was not available in 1970. More women are now going back to work after having children and continuing with their career.
Equal opportunities legislation has helped women get a better deal. More women than men have parttime jobs and they tend to be poorly paid.
Ask learners to look through the solution to this problem. Ask them to comment on it and give opinions on:
 which bits of analysis are not appropriate and why?
 which statistical techniques are appropriate and why?
 are there any other statistical techniques that would have been appropriate?
 comment on the interpretation and conclusion
 how could you improve the solution?
This activity could be adapted to any relevant context using data from the particular sector and adapting the solution presented.
