Modelling (practical approach involving geometry and functions)

Packaging and geometrical shapes

Packaging and geometrical shapes (annotated)
Modelling (practicalapproach involving statistics and functions)

Microwave popcorn

Microwave popcorn (annotated)
Modelling (theoretical approach)

Ice cream

Ice cream (annotated)
Exploring mathematics

Zeno's arrow paradox

Zeno's arrow paradox (annotated)
How long should it be?
It is difficult to be prescriptive about mathematical writing. However, the Mathematics: analysis and approaches and the Mathematics: applications and interpretation guides state that 12 to 20 pages should be appropriate. An exploration may be less than 12 pages, however. A more common failing of mathematical writing is excessive repetition, and this should be avoided as such explorations will be penalized for lack of conciseness. It is recognized however that some explorations will require the use of several diagrams, which may extend them beyond the recommended page limit.
Are there any particular topics to be avoided?
A topic must be chosen so that the assessment criteria can be applied to it. Purely descriptive historical topics, for example, are not appropriate.
Is any particular format for the exploration to be used?
No particular format is required. Students may write both the text of explorations and draw graphs and/or tables by hand, or explorations may be fully or partially wordprocessed. Either form is acceptable as long as the exploration is clearly legible. In recent years, students have used various forms of technology (for example, spreadsheets) to present data, construct tables and graphs, and perform calculations.
Does the exploration need a title?
It is good practice to have a title for all pieces of work. If the exploration is based on a stimulus, it is recommended that that the title not just be the stimulus. Rather, the title should give a better indication of where the stimulus has taken the student. For example, rather than have the title “Number patterns”, the title could be “Number patterns—exploring patterns in final digits of prime numbers”.
What should the target audience be for a student when writing the exploration?
The exploration should be accessible to fellow students.
Can the students use mathematics other than that which they have done in class?
Yes, but this must be clearly explained and referenced, and teacher comments should clarify this.
Can students use mathematics that is outside the syllabus?
It is not necessary to do this to obtain full or high marks. If students decide to explore mathematics outside the syllabus it is recommended that the level is commensurate with the syllabus.
Is interpretation of results a separate section or should comments be made during the exploration?
Commenting on and interpreting results at the point at which these are used enhances the communication and should be summarized in a conclusion. This may also apply to comments on the validity of results.
Must students use external resource material?
There is no requirement for the use of external resource material. However, students often find it necessary to obtain material from other sources (for example, for obtaining data, or for using formulae). In these cases, students should acknowledge these sources and list them in a bibliography and state any sampling processes used when using secondary data.
What is personal engagement?
The exploration is intended to be an opportunity for students to use mathematics to develop an area of interest to them rather than merely to solve a problem set by someone else. CriterionC (personal engagement) will be looking at how well the student is able to demonstrate that he or she has “made the exploration their own” and expressed ideas in an individual way.
What is the difference between precise and correct?
As outlined in criterion E (use of mathematics), “precise” mathematics requires absolute accuracy with appropriate use of notation. “Correct” mathematics may contain the occasional error as long as it does not seriously interfere with the flow of the work or give rise to conclusions or answers that are clearly wrong.
Aesthetics
Calculating beauty–the golden ratio
Colour preferences
Daylight in a classroom–architectural design
Is my mirror showing an accurate image?
M.C. Escher: Symmetry and infinity of art
Modelling the surface area of the glass dome of the Galleria Vittoriio Emanuele II in Milan, Italy
Searching for the ideal sound
Shadows and height
Business and finance
A comparative study of shares, real estate, bonds and banks
Analysis of stock market changes
Applications of calculus to the economics of firms
Buying a car or a house–payment options
Code breaking
Economic development and levels of income
Finding the lowest values of the dimensions of differently shaped storage rooms using differential calculus and optimisation
International phone call pricing
Statistics on flight information for an international airline
Food and drink
Costs of products bought online compared to local grocery stores
Dine in or dine out?
How many peas are there in a 500 gram box of peas?
Jelly bean study
The cookie problem–taste is allimportant
The operation of a tuck shop
The volume of an egg
What is the greatest candy bar in the world?
Health and fitness
A comparison between calorie intake and gender
A comparison between lung capacity, age, weight and body fat
Aids awareness in Maseru
Blood pressure
Breakfast and school grades
Breast and cervical cancer–ethnic comparison
Infant mortality
Investigating reaction times
The SIR model in relation to world epidemics
Geometry and trigonometry
Geodesic domes
Graph theory–finding the shortest path
NewtonRaphson
Origami applications to mathematics
Sine waves in pitch frequencies
Spanning trees
Spherical geometry
Stacking bricks
The ideal cut of a diamond
The Ferris wheel
The open Knight’s Tour on a chessboard
Topography and distance
Nature and natural resources
Airfoil and lift force
Analysis of the cost and utility of gas versus electricity in an average domestic situation
Animal population
Calculating the time of sunrise and sunset
Chaos theory: universal prediction
Counting weeds
Earthquakes–can they be predicted?
Florence Nightingale and modelling spread of disease
Graphing the Pharmacokinetic Profile
How does population density affect the transmission of Ebola?
Is the swell of the sea influenced by the temperature?
Modelling Arctic Sea ice cover
Modelling rainfall
Modelling the cooling of a cup of tea
Optimum dimensions of an aluminium drink can
Predicting cooling times
Rainfall compared to grape vine yield
Statistical investigation of leaves
The quality of local water
The SIR model in relation to world epidemics
The volume of an egg
Sunspot cycles
What is the relationship between the duration of drainage and water height in my bathtub?
Number
Approximation of pi
Cyclic situations and patterns through happy numbers
e,
π
and
: are they related?
The golden number phi
What is e?
Euler’s totient theorem
People
Assuming a person has an 85% chance of meeting a soul mate during their lifetime, what does that mean about the number of potential soul mates in the world?
Correlation between divorce rate and financial uncertainty
Does gender influence choice of favourite animal?
Does the electoral college in the US truly represent the political choice of the people?
Effect on tipping percentages
Exploring the gamblers’ fallacy–why it can cause fatal decisions
Is film genre choice more dependent on nationality or gender?
Genderbased discrimination
Lefthanded students
Memory
Perception of time
Relationship between a country’s human development index and infant mortality rate
Relationship between GDP and fertility rate in countries across the world
Relationship between income inequality and rate of corruption in a country
Relations between international and bilingual students: jobs, pocket money and spending behaviour
Relationship between unemployment and criminality in Sweden from 19881999
Relationship between women’s secondary education and fertility rates in developing countries
Statistical comparison of the number of words in a sentence in different languages
The birthday paradox
When can I use “swimmed” and “knowed” correctly?
Voter turnout
Sport and leisure
Baseball bat speed compared with body weight
Body proportions for track and field events
Does the team win when it was the dominating team during the match?
Effective short corners in hockey
Exploring card counting in blackjack using probability
Factors affecting athletic performance
Has sports performance improved more on land or in water?
Height, weight and swimming performance
How does the amplitude of a ski turn affect the speed of the skier?
How far do tennis balls roll?
The geometry involved in billiards
Modelling musical chords
Modelling the jump of a horse
Practice makes perfect
Relationship between skiing ability and distance travelled to ski
Resistance of fishing line
Rollerblading and the maths behind it
The Monty Hall problem
The Tower of Hanoi puzzle
Video games and response times
Will female swimmers ever overtake male swimmers?
Travel and transport
Cost efficiency of vehicles
Driving skills
How many bicycles are there in Amsterdam?
Petrol prices
Public transportation costs and car usage: a personal comparison
Running late and driving habits
Seat belt use
The effect of blood alcohol content law on the number of traffic collisions in Sacramento
Traffic study of Schiphol International Airport
Transport safety in town centres