Modelling Motion and Gradient Constraints in a Custom Roulette Game MMT3-IA1

Mathematical Methods

Conditions

Technique Problem-solving and modelling task

Unit Unit 3: Further calculus

Topic/s Topic 2: Further differentiation and applications 2

Topic 3: Integrals

Duration 4 weeks (including 3 hours of class time)

Mode / length Written: Up to 10 pages (including tables, figures and diagrams) and a maximum of 2000 words

Individual / group A unique response must be developed by each student

Other Use of technology is required and must go beyond simple computation or word processing

Context

Humans have for long been drawn to games. Some people even like to take risks with games of chance. Casinos are sites where people try to win money playing games of chance. The odds of winning are determined by the casino, although often these odds do not match with the 'real' odds. According to long run probability, the casino always wins. This applies to most gambling (https://www.gambleaware.nsw.gov.au/learn-about- gambling/what-are-the-odds).

An example of a such a game is roulette.

The game allows for myriad ways of gambling (and losing!) including betting on single numbers from 1 to 36 (that includes a 0 or 00 to increase the likelihood of a win for the casino in the long run).

A casino wants to develop a new type of game that is similar to roulette. The game consists of two ramps (that terminate at A and B) and a wheel that rotates at a constant velocity in an anti-clockwise direction. This wheel is composed of 20 adjacent and equally-sized sectors (slots). Five slots are shown from above (aerial view) below.

A rubber disc is released from a chute above the centre of the wheel so that it slides down one of two ramps and lands in one of the slots. A basic set up is shown below.

Task

You have been issued with the task of extending the design given above so that the ramps are more complex and not merely straight. You are issued with some parameters in the Stimulus section which relate to the diameter and rotational speed of the wheel and the maximum height of the game.

Incorporating topics covered in Unit 3 relating to various types of functions and their derivatives and integrals, you are to take the following into account:

• the ramp contains at least 3, but no more than 5, of logarithmic, exponential, trigonometric and polynomial Straight segments are allowed but do not contribute to the number of functions. Complexity must be considered.

• All joints along the ramp are smooth so that the disc travels without jumping/skipping.

• The disc should not leave the surface of the ramp while it is Prior testing has demonstrated that this can be assured if the maximum gradient along the ramp is 70.

• The casino wishes to leave a rectangular section to insert a name for the game/logo. They want this area to be centred on the triangle and to be 1/3 of the cross- sectional area. You are to place this rectangular section and state its area.

• Develop and compare equations and graphs that describe the motion of discs that land at A and B as they travel through one rotation.

• Casino employees have debated whether the frequency of discs leaving the chute should be constant or Discuss the reasonableness of these, incorporating your equations and/or graphs from the step above.

To complete this task, you must:

• use the problem-solving and mathematical modelling approach to develop your response

• respond with a range of understanding and skills, such as using mathematical language, appropriate calculations, tables of data, graphs and diagrams

• provide a response that highlights the real-life application of mathematics

• respond using a written report format that can be read and interpreted independently of the instrument task sheet

• develop a unique response

• use both analytic procedures and

Stimulus

Diameter of wheel from middle of opposing sectors: cm

Rotational velocity of wheel: rev/s

Maximum height of game: cm

Checkpoints

One week after issue date: Students email progress to their teacher (Design plan, identify procedures).

Two weeks after issue date: Complete mathematical calculations of investigation, refine model technology and algebraic solutions evident. Start evaluation.

Three weeks after issue date: Evaluate & verify model. Submit draft for individual feedback from teacher.

Four weeks after issue date: Students submit their final response.

Authentication strategies

• You will be provided with three hours of class time towards task

• Students' progress will be documented and copies of student responses collected at the checkpoints

• Your teacher will ensure class cross-marking occurs where possible

• Each student will produce a unique response using individualised data

• By submitting your response, you agree to the following:

I declare that all work contained in this assessment submission is entirely my own and was completed in accordance with the task conditions and any relevant AARA support provision. Where applicable, I have included all planning, drafting and other required documentation with my final submission.

Instrument-specific marking guide (IA1): Problem-solving and modelling task (20%)

Criterion: Formulate

Assessment objectives

1. select , recall and use facts, rules, definitions and procedures drawn from Unit 3 Topics 2 and/or 3

1. comprehend mathematical concepts and techniques drawn from Unit 3 Topics 2 and/or 3

1. justify procedures and decisions by explaining mathematical reasoning

Criterion: Solve

Assessment objectives

1. select , recall and use facts, rules, definitions and procedures drawn from Unit 3 Topics 2 and/or 3

1. solve problems by applying mathematical concepts and techniques drawn from Unit 3 Topics 2 and/or

Criterion: Evaluate and verify

Assessment objectives

4. evaluate the reasonableness of solutions

4. justify procedures and decisions by explaining mathematical reasoning

Criterion: Communicate

Assessment objectives

3. communicate using mathematical, statistical and everyday language and conventions