Nuffield Mathematics teaching resources are for use in secondary and further education

FSMQ Level 3 (pilot) Mathematical principles for personal finance scheme of work

Before starting this Advanced (Level 3) FSMQ, students should have acquired the skills and knowledge associated with the Number and Algebra assessment objective of GCSE Mathematics, or with an Intermediate (Level 2) FSMQ Finance course, or equivalent.

A suggested work scheme showing topics and methods to be covered is given below.  The order in which the topics are covered and the time allocated to them can be varied to suit different groups of students. 

Topic area

Content

Nuffield resources
The links below go to pages from which you can download the resources, some recently revised.

Percentages and personal taxation
(3 hours)

Calculate

% change = \frac{\textup{current~value - previous~value}}{\textup{previous~value}}  x 100

Use multipliers to combine % changes (including repeated changes) and to reverse % changes - including VAT.

Carry out complex calculations involving multiple rates – including income tax, national insurance, capital gains tax.

Working with percentages   
Discussion and worksheets supported by presentation – check what methods students use for some real-life percentage problems and introduce/ revise the use of multipliers.

Income Tax  
Notes, examples and exercises based on rates for the current tax year.

National Insurance  
Notes, examples and exercise based on rates for the current tax year.

Saving
(4 hours)

Class discussion - Would it be better to receive a sum of money (such as £5000) now or at a later date (such as 20 years from now)? 

Discuss the key idea of present value (PV) and future value (FV).

Collect and discuss information about interest rates from banks, building societies and other sources.

(Include information given in the form of tables and diagrams.)

Use ideas of compound interest to introduce the formula FV = PV(1 + r)n where r is the interest rate expressed as a decimal and n is the number of time periods. 

Apply recurrence relations such as  P_n_+_1 = P_n(1 + r) .

Also transform to PV =\frac{FV}{(1 + r)^n} and apply this to find present values. 

Include the use of both a calculator and spreadsheet (if possible).

AER: Calculate the annual effective interest rate, r, given a nominal interest rate, i

r = \left ( 1 + \frac{i}{n} \right )^n - 1  where n is the number of compounding periods per year.

 

Savings growth  
Leaflet giving interest rates on three savings accounts and worksheet based on the leaflet. This introduces the idea of a recurrence relation and includes the use of a graphic calculator, Excel spreadsheet and formulae to find future values and the AER.

 

Topic area

Content

Nuffield resources
The links below go to pages from which you can download the resources, some recently revised.

Indices
(5 hours)

Use data relevant to students' other studies or interests to introduce the idea of an index as a ratio that describes the relative change in a variable (e.g. price) compared to a certain base period (such as one year).  Include examples from the National Statistics website at www.statistics.gov.uk (such as indices for prices, earnings, building costs, manufacturing output, motor vehicle production, and retail sales volumes).

Include % change = \frac {\textup{current~index} - \textup{previous~index}}{\textup{previous~index}} \times 100

for areas of finance such as the FTSE 100 share index.

Research and compare the Retail Price Index (RPI) and Consumer Price Index (CPI).

Include calculation from the Laspeyres Index formula (weighted by quantities in the base period): 

I_L = \frac{\Sigma P_i_tQ_i_0}{\Sigma P_i_0Q_i_0} x 100

where  Pit is the price of commodity i at time t,  

and      Qit is the quantity of commodity i at time t.

0 represents the base period, so for example Qi0 represents the quantity of commodity i at the base period ( t = 0 )

Average earnings index  
Learners use average earning index numbers to compare the way in which earnings in different sectors have changed over different time periods.

Inflation indices   
Activity in which students find information about RPI and CPI and answer questions – will provide a set of notes.

Laspeyres Index  
This activity uses a simple coffee shop example to introduce the Laspeyres index formula. It gives learners practice in real contexts, then asks them to calculate a Laspeyres index for their own regular spending pattern.  (Includes optional use of the Personal Inflation Calculator on the National Statistics website).

Data over time
(4 hours)

Inspect tables and graphs considering changes over time.  Include data over different time intervals (such as daily, weekly, quarterly).  Interpret trends (such as in stocks & shares, interest rates, exchange rates).  Represent data graphically and find linear equations to model data using gradient and intercept and algebraic substitution.

Calculate average changes.  Use moving averages to smooth short-term fluctuations and interpret situations (eg involving indices such as the 100 share index).  Include seasonal and cyclical variations.

Moving averages:
For data points p_1, p_2, ... the simple moving average at interval m with n data points is

\overline {x_m} = \frac{p_m + p_m_-_1 + p_m_-_2 + ......p_m _-_(_n_-_1_)}{n}

Calculate successive values using

\overline{x_m _+ _1} = \overline{x_m} - \frac{p_m _- _(_n _- _1_)} {n} + \frac {p_m_+_1}{n}

Address the problem of lag using weighting and calculate the linear weighted moving average

where the denominator is the triangular number with sum  \frac {n(n + 1)}{2}

Understand that this can be considered to lag behind the trend.

House prices moving averages  
Uses house prices to introduce moving averages and weighted moving averages. Includes spreadsheet giving data for regions of the UK. Learners can use this to compare what has happened in their local area with the national picture or other regions.

 

Topic area

Content

Nuffield resources
The links below go to pages from which you can download the resources, some recently revised.

Borrowing
(3 hours)

APR (annual percentage rate)
Use the simplified version formula for APR for a single loan repaid in full after a fixed period:
Overall cost, C = \frac {A}{(1 + i)^n} where n = number of years between start of loan and its repayment.

APR – Annual Percentage Rate  
Introduces the formula for APR that applies when a single loan is repaid in full after a fixed period and gives learners practice in using it.

Financial diagrams
(3 hours)

Make sense of a range of information presented in tables and diagrams relating to finance.

Include tabulated information about savings, loans, credit card accounts, shares, ISAs (etc.) and information in the form of graphs and charts.

Financial graphs and charts  
Links to useful websites where up-to-date tables and diagrams can be found.

Weighted averages
(4 hours)

Calculate contributions made by individual items to indices. For instance calculate contributions made by the prices of commodities in different shops and regions to a consumer price index. 

For example, if a commodity costs £5 in shop A and £6 in shop B and 40% (0.4) of customers buy the commodity from shop A and 60% (0.6) from shop B, the effective cost of the commodity used in calculating an index is 0.4 × £5 + 0.6 × £6 = £5.60

Calculate a composite index by combining indices using weighting (for instance in calculating a price index the index of each commodity multiplied by its weighting is totalled, and this sum is divided by the sum of all the commodities weights).

 

Further borrowing
(8 hours)

Use more complex recurrence relations such as for credit cards, mortgages.

Find APR (annual percentage rate) for a loan repaid in a small number of instalments (such as 2, 3 or 4).

using loan:

C = \frac{A_1}{1 + i} + \frac{A_2}{(1 + i)^2} + \frac{A_3}{(1 + i)^3} + \frac{A_4}{(1 + i)^4}

· substituting values into the above equation for confirmation,

· solving the above equation for i using the interval bisection method.

Include applications to financial areas such as loans, credit cards, mortgages.

Credit cards  
Learners use recurrence relations to work out how long it takes to pay off credit card debts.  Includes the use of both a graphic calculator and spreadsheet.

APR with more than one instalment  
How to find and check APR values in cases when a loan is paid back in more than one instalment. Includes plenty of practice in using the interval bisection method and with a spreadsheet set up to use this method to calculate APRs.

 

Topic area

Content

Nuffield resources
The links below go to pages from which you can download the resources, some recently revised.

Further indices
(8 hours)

Use the Paasche index formula (weighted by quantities in the calculation period):

I_P = \frac{\sum P_i_tQ_i_t}{\sum P_i_0Q_i_t} \times 100

 

where  Pit is the price of commodity i at time t,
and      Qit is the quantity of commodity i at time t


Research and compare fixed base and chain linked indices: to develop an understanding that in a chain index comparisons are always made between subsequent points and therefore take account of changes between the start and end points


 Use the Fischer index formula I_F = \sqrt{I_L \times I_P}
 
(geometric mean of Laspeyres and Paasche indices).

Compare Laspeyres, Paasche and Fischer indices.

 

More tables and diagrams
(6 hours)

Use information given in more complex tables relating to personal finance.

Discuss and make sense of a range of less familiar charts eg hi-low charts, candlestick charts, Kagi charts.

Financial charts and graphs   
Links to websites with up-to-date tables and diagrams.

Continuous compounding
(4 hours)

Introduce and use the idea that continuous compounding leads to exponential functions. That is: 

P = P_0\left(1 + \frac{r}{n} \right )^n^t

 is the amount after t years for an initial investment of P_\mathit{0}  when the interest is compounded n times per year, and n \rightarrow \infty 
giving P = P_0e^r^t

 

Revision
(8 hours)

Revise topics.  Work through revision questions and practice papers.

Discuss the data sheet - make up and work through questions based on it.

Financial calculations  
A set of worked examples to help with revision.

 

 

Page last updated on 25 January 2012