MEI Conference 2014
IQM: Modelling and
estimation
Stella Dudzic
stella.dudzic@mei.org.uk
MEI Introduction to Quantitative Methods
Roughly Speaking
Questions: calculating with error bounds
1 An archaeologist is investigating the remains of an Anglo-Saxon settlement which burnt down.
She uses three different techniques for dating the fire in the settlement. The results are in this
table.
Object dated Estimated age Error bound
Wooden posts 1200 years
± 100 years
Coins
1280 years
± 30 years
Pottery
1210 years
± 50 years
(a)
The archaeologist sketches the estimate for how many years ago the fire was based on the
age of the wooden posts on a number line, like this.
1100
1200
1300
Add the other two estimates to the archaeologist’s diagram.
(b)
What can the archaeologist deduce about the age of the settlement?
2 A few miles from the settlement the archaeologist uncovers an iron-age burial plot. She estimates
that the human remains are 2400 ± 100 years old.
(a)
From a complete skeleton the archaeologist estimates the height of an iron-age woman as
140 cm ± 3 cm. Write this information as an inequality.
(b)
From a few bones the archaeologist estimates the height of an iron-age man as 150 cm,
correct to the nearest 10 cm.
Is there a larger possible percentage error in the estimate of the man’s or woman’s height?
(c)
Near the burial plot the archaeologist finds a piece of pottery in a style thought to be first
made around 300 BC. Could the pottery have belonged to the people buried in the grave?
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Estimation
1
Songs by my favourite band cost either 79 p or 99 p to download. I have a £10 gift card. What can
I afford?
2
It takes me 15 – 20 minutes to walk a mile in town. How far should I plan to walk between
campsites on my walking holiday in the lake district?
3
There are 20 people ahead of me in the queue for train tickets. 3 people have got tickets in the last
5 minutes. How much longer until I am at the front of the queue?
4
A train ticket to London costs £37.50 return. I travel to London 3 days a week. Is it worth getting
a £600 monthly ticket?
5
I have an electricity smart meter in my home. It tells me that yesterday I spent £2.64 on
electricity. How much will my bill be for a year?
6
A group of 8 friends is out for the evening in a nearby town. A 5-seater taxi will charge them £27
for the journey home. Roughly how much each will the fare home cost?
The modelling cycle
Specify a problem
Make assumptions to enable
representing the problem in
mathematical form
Review
assumptions
No
Is the solution
satisfactory?
Yes
Present findings
Solve to produce
theoretical results,
interpreting them in the
context of the problem
Select information from
experiment, experience or
observation and compare
with theoretical results
MEI Introduction to Quantitative Methods
Financial Problem Solving
Section 3: Financial problem solving
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MEI Introduction to Quantitative Methods
Working with Exponentials
Section 2: Exponential growth and decay
Simulating radioactive decay.
Materials: at least 100 small cubes with one side painted black
If you don’t have cubes you can do this with coins or dice.
The decay of a radioactive material is completely random and can take a very long time.
However, scientists can predict how many radioactive nuclei will decay in a given amount of
time – they just don’t know which ones will decay first.
An experiment to simulate the decay of a radioactive material can be performed using a
large set of cubes. One face of the cube is painted black. The cubes are thrown and the
ones that land ‘black side up’ are considered to have decayed and are removed. This
simulates the radioactive decay of an unstable material.
If you are using coins, then the ones that land on a head are considered to have decayed.
If you are using dice, then the ones that land on a 6 are considered to have decayed.
The time interval for half the total number of nuclei in a material to decay is known as the
radioactive half-life of the material. This experiment seeks to establish the length of the halflife of a material. We shall also find an equation that can predict the number of remaining
nuclei at any time.
Split up into groups and share the cubes (coins) between the groups.
Groups should gently throw the cubes on to the bench and then count the cubes which land
black side upwards and move them to one side (or the coins which land head up).
Have a recording volunteer who asks each group in turn how many cubes ‘decayed’ and
adds these up to give a total score for throw number one.
All the scores are then recorded in a table of three columns shown below.
Gather together the remaining, undecayed cubes and repeat the process. The process is
repeated until there are no more cubes left to throw.
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Number Throw
(x)
Number of
decayed nuclei
(counters removed)
Number of stable nuclei
(counters remaining)
(y)
0
1
2
3
4
5
6
To consider:
Draw the graph of your results on excel on a scatter graph.
What is the probability of each cube landing on the decayed face?
Can you write up a formula for what you expected to happen?
Compare your results to the expected results.
What do you think would happen if you used a large number of cubes?
You can use the excel spreadsheet called simulating cubes to do this for up to 1000 cubes.
If you are using coins you can use http://shodor.org/interactivate/activities/Coin/ to simulate a
large number of coins.
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