The formulae to calculate the area of this cross section are given in Appendix 1.
Triangular cross section(V shape)
-
Area = x () x () sinθ
A =w2 sinθ
By varying θ by 1°, where 0° is at the point of the two sides touching each other to make a straight line. Below is a snapshot of the excel spreadsheet showing optimum results.
The formulae used to calculate the area of this cross section is given in appendix 2.
By using algebra I can find θ at which the optimum capacity exists.
Angle to give Optimum capacity;
=w2 sin θ = 0
Cos θ = 0
θ = 90°
Semi-Circular cross-section
l = θ x r
l = θ x r
r =
∴ r =
Area of semi circle = ½ x pi x r2
Formulae used in excel for this calculation is given in Appendix 2.
By the use of conditional formatting of the entire spreadsheet I have found that the best cross section to use for a gutter to carry the maximum water capacity is a semi circle.
2)
Using Excel; it will take 21 months to clear the debt. Of which that last month is a small payment under £70.
Total interest paid in this case = £201.89
To clear the debt off in half the time the payments will have to increase to approximately £130. This is an increase of approximately 85% of the original monthly payment.
To clear off the debt in double the time the payments will be lower, decreasing to around £40. Approximately 40% lower.
A copy of the spreadsheet is given overleaf followed by the formulae used to create this spread sheet. As a little extra this spreadsheet allows a user to enter a period of time, in months, to clear the debt and the monthly payments required will be calculated.
N.B.
The other additional extra which uses the monthly payment will calculate the time, in months; it takes to clear the debt and displays the total interest payable. This is created by using a different formula and because of that it displays the total interest to be 2 pence lower that the other method. The reason for this is that the formula used has calculated a part month interest, where the initial method takes the interest as a full month.
3) Excel has a variety of uses. One benefit of using spreadsheets for both question 1 and 2 is speed. For instance question 1 is to determine best possible design of a gutter. By the use of formula the volume of a design is quickly calculated. Replication of formulae enables the effortless generation of a large number of calculated values across a chosen range of values (Mathematical Association, 2002, p39). By calculating the formula for the first two rows in Appendix 1 then dragging these formulae down the cells uses the formulae with the corresponding values to enable fast calculation of the area.
By manipulating the formulae the volume of the other designs can be generated. This is a lot quicker in excel compared to using pen, paper and calculator. It also allows the recording of all calculations. A spreadsheet is instantly responsive to changed input values which enables exploration of the effect of variables within a process (Mathematical Association, 2002, p39)
With the conditional formatting function in excel it is easy to locate which cross section gives the highest value. This saves time comparing manually as the spreadsheet finds the greatest value automatically. The Mathematical Association (2002 p39) state “value sensitive formatting allows the appearance of text, values, and background, to change according to the intentions of the designer, and so give high value feedback or other information, to the user”
Using a spreadsheet in question 2 allows interactivity. Linked values are a defining feature of the spreadsheet. This capability to link and respond enables the spreadsheet to offer interactivity (Mathematical Association, 2002, p39).
The benefit of the replication of formulae is also a benefit to using a spreadsheet in question 2. The first two rows were calculated first using the formula given in the table on the previous page. By filling down with this formula, the other rows contain the same calculation for the corresponding cells which calculate a monthly figure reducing the debt and taking into account the interest rate.
4) Activity for key stage 3.
Question 1) The bar chart below shows the amount of money collected by 100 pupils on a sponsored walk. (Perks and Prestage, 2001b, p94)
a) Estimate the mean amount collected per child.
b) What is the mode?
Now using the table in figure 1, that created the above graph, change the frequencies so that the mean is 6.
Figure 1: Excel spreadsheet of a frequency table
Strategies used by pupils will vary. Some may use symmetry, some just trial and improvement, others may think about patterns of numbers.
Extension work: Can the mean be 5? Can you get the mean to be between 6 and 7?
- Obtain a mean of 5 and mode of 4
- Obtain a mean of 6 and mode of 3.
Pupils will firstly learn to read frequency from a bar chart and to calculate the mean themselves. They will have to know the formula for the mean to answer the question. This will make the pupil think which formula needs to be used.
The mode can be seen from the bar chart. As the numbers in the frequency column change the pupils will be able to see the mean being recalculated and the bar chart adjusting itself to the new values. This allows the pupils to see the effect of manipulating data has on the reproduction of bar charts. Prestage and Perks (2001a, p60) suggest as the solutions can be found by trial and improvement, there needs to be an expectation that pupils explain their results; otherwise the mathematics of the task will be diminished.
Question 2)
(Perks and Prestage, 2001b, p95)
a) Change the raw data in the spreadsheet to give a mean of 6.
b) Get a mean of 5 and a mode of 4.
c) Get a mean, median and mode of 5.
d) Get a mean of 4, a median of 3 and mode of 5
Question 3) (. (Perks and Prestage, 2001a, p62)
An article in the Evening Post reads ‘The average sponsorship money collected by each child was £6.00; what an effort!’ The Evening News stated ‘An average of £5.52 – what a collection!’ Both papers showed the same diagram. Which paper got its sums right?
In this question the pupil would use the original bar chart from the 1st question to find the averages. There is however no indication of whether mean, median or mode is expected. Both newspapers could have the correct information but choose to display it in a different way. The mathematics focuses on understanding the connection between raw data and the smoothing of the data shown in representation. (Prestage and Perks, 2001a, p62)
References
Math Forum, 2003, Maximizing The Volume of a Rain gutter. [online] Available at <http://mathforum.org/library/drmath/view/64541.html> [Accessed 03 November 2010]
Perks, P., and Prestage, S., 2001a. Adapting and Extending Secondary Mathematics Activities. London: David Fulton Publishers
Perks, P., and Prestage, S., 2001b. Teaching the National Numeracy Strategy at Key Stage 3. London: David Fulton Publishers
Vertex42, 2003-2010, Debt Reduction Calculators. [online] <http://www.vertex42.com/Calculators/credit-card-payoff-calculator.html> [Accessed 03 November 2010]
The Mathematics Association, 2002. ICT and Mathematics: a guide to learning and teaching mathematics 11-19, Produced for the Teacher Training Agency.
