Download Teaching Innovations in Business Mathematics II

Teaching Innovations in Business Mathematics and Statistics
Dr. Oded Tal, Conestoga College
1. Introduction
This paper presents two teaching innovations which have been proven to contribute to
students’ understanding and success. The first innovation is using a saw-tooth model to
explain perpetuities in Business Mathematics II. The second innovation is a simpler
method of applying the Continuity Correction Factor to the normal approximation to the
binomial distribution in Statistics.
2. Using a saw-tooth model for teaching perpetuities in Business Mathematics II
A saw-tooth model is often used in science and engineering to represent a continuous
process, which fluctuates between two constant levels at regular intervals. A typical
example is used in inventory management, in order to generate automatic optimal size
orders, based on past consumption statistics and on order lead time (Appendix 1).
In Business Mathematics a modified saw-tooth model is a powerful visual aid for
teaching the perpetuity principle. In this case the vertical axis represents the money in the
bank, and the tooth width is the payment period, usually one year. At t=0 the money in
the bank is the original investment, PV. At the end of each payment period a simple
interest PV*i is added, and an annual payment PMT is immediately subtracted. There can
be three different situations:
Situation no. 1: PMT>PV*i (Figure 1): The original funds will be gradually depleted,
which means that it is an annuity, rather than a perpetuity, and that the initial investment,
PV, was too small.
Situation no. 2: PMT<PV*i (Figure 2): The money in the bank will steadily grow,
which means that the initial investment, PV, was too large.
Situation no. 3: PMT=PV*I (Figure 3): The money in the bank will forever fluctuate
between PV(1+i) and PV. This means that the original investment, PV, was the right
amount for a perpetuity. Since PMT=PV*i, then the required amount is given by
PV=PMT/i.
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Money in
the bank
PV*i
PV
PMT
0
0
1
2
3
4
5
Years
Figure 1: PMT>PV*i
Money in
the bank
PMT
PV*i
PV
0
0
1
2
3
Figure 2: PMT<PV*i
2
4
5
Years
Money in
the bank
PV*i
PMT
PV
0
0
1
2
3
Figure 3: PMT=PV*i
3
4
5
Years
3. A simpler method for applying the continuity correction factor
When using the normal approximation to the binomial distribution in Statistics a
continuity correction factor of 0.5 has to either be added or be subtracted from the value
of interest, X. A common way of explaining how to apply the correction factor asks the
student to distinguish between 4 cases (e.g. ref. 1-3) or even 6 cases [4], and to memorize
what to do in each case, which may be quite confusing. For example:

For the probability of at least X occur, use the area above (X-0.5)

For the probability that more than X occur, use the area above (X+0.5)

For the probability that X or fewer occur, use the area below (X+0.5)

For the probability that fewer than X occur, use the area below (X-5)
The following 3-step method has proven to be easier to understand and to remember:
Applying the Continuity Correction Factor
Step 1: Rephrase the question using one of the following conditions (corresponding to
the above categories):
1.
P(X>z)
2.
P(X>z)
3.
P(X<z)
4.
P(X<z)
Step 2: In case of conditions of type 2 or 4 transform the original condition to an
equivalent condition of type 1 or 3 respectively by figuring out the smallest or the largest
integer that satisfies the original condition. For example: P(X>5) is equivalent to P(X<6);
P(X<9) is equivalent to P(X<8).
Step 3: Draw a normal distribution graph and shade the required area, and then increase
the shaded area by either adding or subtracting 0.5.
Dual parameters conditions can also be easily solved using the same method, as
demonstrated by the following three examples.
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Example no. 1: P(8>X>3)
Step no.3: After drawing the normal distribution graph, increase the shaded area at
both ends, which means that the new points of interest become 8.5 and 2.5 (Figure 4).
Example no. 2: P(8>X>3)
Step no. 2: Transform P(8>X>3) to the equivalent condition P(7>X>4).
Step no.3: After drawing the normal distribution graph, increase the shaded area at
both ends, which means that the new points of interest become 7.5 and 3.5.
Example no. 3: P(8>X>3)
Step no. 2: Transform P(8>X>3) to the equivalent condition P(7>X>3).
Step no.3: After drawing the normal distribution graph, increase the shaded area at
both ends, which means that the new points of interest become 7.5 and 2.5.
i
t
i
:

,
.
f
.
-
2.5
3
8
Figure 4
5
7.5
References
1.
Lind, D.A., Marchal. W.G., Wathen, S.A and Waite, C.A., Basic Statistics for
Business and Economics, second Canadian edition. Mc-Graw-Hill Ryerson,
2006.
2.
Bluman, A.G. , Elementary Statistics- a brief version, 3rd edition. McGraw-Hill,
2006.
3.
Triola, M.F., Goodman, W.M. and Law, R., Elementary Statistics, second
Canadian edition. Addison Wesley Longman, 2002.
4.
Black, K., Business Statistics for Contemporary Decision Making, 4th edition.
Wiley 2004.
5.
http://www.future-fab.com/documents.asp?grID=384&d_ID=3009.
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Appendix I: An inventory management saw-tooth model [5]
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