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FE Magazine - Issue 37: Functional Maths


Created on 14 August 2014 by ncetm_administrator
Updated on 27 August 2014 by ncetm_administrator

 

FE Magazine - Issue 37'Chocolate Truffles' by Au Kirk (adapted), some rights reserved
 

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 self-critical. 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 self-assessing 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 made-up 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 square-based pyramid shape.

Volume of each chocolate = πr2h = π × 0.75 × 0.75 × 1
  = 1.34 cm3
   
The volume of space needed for each chocolate = 1.5 × 1.5 × 1 cm3
  = 2.25 cm3

Arrange them like:

Bottom layer Second layer
diagram showing 16 chocolates diagram showing 9 chocolates
Third layer Fourth layer Top layer
diagram showing 4 chocolates diagram showing one chocolate diagram showing one chocolate

From the top it will look like:

diagram showing chocolates in pyramid viewed from above

From the side it will look like:

diagram showing pyramid of chocolates viewed from side

Choose dimensions of triangle to be:

base: 8cm

height: 7cm

diagram showing pyramid of chocolates in box viewed from side

Using Pythagoras: 82 + 3.52 = 76.25

√76.25 = 8.73 cm

This is the net of my final design:

net of final design

Area of triangle = ½ x 8 x 7 = 28cm2

Area of square = 8 x 8 = 64cm2

Area of card needed = 92cm2

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:

bar chart showing wages per year

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:

pie chard showing percentages and their occurences

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

line graph showing trend over time

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

previous chart redrawn with different axes

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 part-time 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.

Image credits
Page header by Au Kirk (adapted), some rights reserved

 

 
 
 
 
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Comments

 


28 August 2014 13:04
Thank you for your comment. You are of course quite correct.

These are 'made up' solutions which are not necessarily the best or most accurate solutions. The idea is to give them to learners to look at critically and analyse to spot any errors, mistakes and/or misconceptions and to make improvements to lead to better solutions.

Please refer back to the beginning of the article for more information/guidance .....

"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 self-critical ......."
By vivbrown
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28 August 2014 10:45
Love problem 1, but aren't there 31 chocolates in the possible solution?
By cathmoore
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