LOGIC DIAGRAMS
A. Network Diagrams
Students should be given a note on what a network diagram is.
E.g. ‘A network diagram is simply a series of points (nodes or vertices) joined by a series of
lines or curves (arcs).’
The introductory question should be explained. This exercise should be attempted according
to the four rules (a) to (d).
Define traversability.
(Note: this network cannot be drawn and the reason why it can’t
will be developed during the next two exercises).
Exercise 1 (question 1 only), may now be attempted. Students
should be encouraged to draw all the diagrams in exercise 1
neatly and to keep them safe for later use. The 18 diagrams should be sorted into two sets,
those which can be drawn and those which can’t.
Definition: The order of a node is the number of arcs which enter (or exit) it.
Odd and even nodes should be defined.
Exercise 1 (questions 2–6), may now be attempted. The results of exercise 1 should now be
summarised. It should be pointed out to students that in each diagram all of the nodes were
even or all of them were odd.
Exercise 2 (questions 1–5), may now be attempted. The results of exercises 1 and 2 should
now be summarised and students allowed to try the Königsberg Bridges Question.
B. Tree Diagrams
Students should be shown this simple tree diagram drawn to show all the possibilities when a
coin is tossed twice. Students should be shown how to list all the possibilities from the tree
diagram.
It should be stressed that tree diagrams are useful because they illustrate all the possibilities.
head
head
tail
head
tail
tail
HH
HT
TH
TT
Exercise 3 may now be attempted.
Mathematics: Applications of Mathematics (Int 2) – Staff Notes
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C. Shortest Path Problems
This topic could be introduced using the
following example:
A taxi driver picks up three passengers from a
party in Gogor.One lives in Hagton. One lives
in Isley and the other lives in Fairlee. Which
way should the taxi driver go so that the
distance covered is a minimum?
Hagton
Gogor
7 km
F
G
7
F
9
9
6
I
10
H
Isley
Total = 23 km
F
Total = 23 km
6
I
9
I
9 km
Total = 27 km
I
H
8
6 km
10 km
Fairlee
There are several ways of listing all the
routes, but a tree diagram is the most useful
10
8 km
Total = 22 km
6
H
Shortest route is Gogor —> Fairlee —> Isley —> Hagton (22 km)
Exercise 4 (questions 1 – 7), may now be attempted.
Ay
to
n
Beeton
16
11
20
16
Ce
15
17
18
Ayton
et
ow
n
11
Be
et
o
n
17
20
Ceetown
15
D
ee
to
n
18
Deeton
Show students how to draw a map from a Distance Table and vice-versa.
Exercise 4 (questions 8 – end), may now be attempted.
Mathematics: Applications of Mathematics (Int 2) – Staff Notes
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D. Flowcharts
Students will probably have met flowcharts at various stages of their education.
Show the various types of boxes
START
Decision
boxes
Calculation
boxes
STOP
START
Yes
Are you
over 16 years
old?
Fare = £1·80
No
Fare = £1·20
STOP
This is a simple flowchart showing how fares depend on whether you are a child or an adult.
Exercises 5 and 6 may now be attempted.
Mathematics: Applications of Mathematics (Int 2) – Staff Notes
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E. Spreadsheets
A spreadsheet and its uses should be defined.
E.g. ‘A useful table for keeping a note of names and numbers and for doing multiple
calculations’.
This part of the work should ideally be done using a computer suite. If there is a timetabling
problem, it could be completed over one to two periods per week running in tandem with the
rest of the unit.
It could also be worked on (manually!) without access to a computer, but even here, it is
recommended that students see at least one of their spreadsheets live in action on a computer,
either an early one or a later one.
The notes were made up with the spreadsheet in Claris 4 in mind, but any Applemac or P.C.
computer spreadsheet is just as good.
Blank grids are given on the next page if access to a computer is limited and can be easily
photocopied for the students.
Note that it may be that some students have met spreadsheets before.
Mathematics: Applications of Mathematics (Int 2) – Staff Notes
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Manual Spreadsheeting Blanks
A
B
C
A
1
2
3
4
5
6
7
8
9
10
B
C
D
1
2
3
4
5
6
7
8
9
10
A
B
C
D
E
F
1
2
3
4
5
6
7
8
9
10
11
12
13
A
B
C
D
E
F
G
1
2
3
4
5
6
7
8
9
10
11
12
13
14
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The layout of a spreadsheet should be explained, i.e. the cells, cell position (B13) and the
menu bar.
Students can be shown how to click into cells and insert names and numbers. They should be
shown how to highlight cells in order that fonts, sizes, styles and alignment can be altered
using the Format and Edit commands in the menu bar.
They should be given a blank disc and shown and encouraged to save their spreadsheets for
use in later exercises.
A naming system is suggested, however students could devise their own.
Exercise 7 may now be attempted.
Students should be shown how to multiply, divide, add and subtract the contents of cells.
Example:
If you wish the sum of the contents of cells A3, B3 and C3 to go into cell D3
(a) Tap on the cell where you wish the answer to go (D3).
(b) Type in ‘ =’ (you can see it appearing just under the menu bar).
(c) Type in A3+B3+C3 (return) – the answer appears automatically!
Students should be shown how to copy down a formula from one cell to a series of cells
below it.
Example:
If you wish your above formula in cell D3 to be copied down to cells D4 to D8
(a) Drag your cursor from cell D3 to D8 (all the cells will darken).
(b) Go to ‘Copy Down’ in the Calculate command in the menu bar.
(c) Check that the formulae are correct
Exercise 8 may now be attempted.
Students should be shown some of the shortcuts SUM, AVERAGE which appear in
the‘Paste Function’ in the Edit command in the menu bar.
Students should be allowed to view and try out some of the functions available.
Example:
The SUM function. If you wish the sum of the numbers in cells A3 to C3 to go into cell D3.
(a) tap on cell D3 and type in =
(b) go to ‘SUM’ in the Paste Function under the Edit command in menu bar.
(c) go to the top of the page and highlight the contents of the brackets.
SUM( number1;number2;... )
(d) Draw the cursor across cells A3 to C3.
SUM(A3..C3) now appears and gives the answer.
Exercise 9 may now be attempted.
Mathematics: Applications of Mathematics (Int 2) – Staff Notes
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