*NMBR140/ACHE 105 Introduction to Mathematical Thinking*

*ASSESSMENT TASK 1 (AT1): PROBLEM SHEET *

You are to *attempt* all 4 problems giving written fully justified mathematical arguments for your solutions. All strategies used should be fully documented. Use a variety of resources such as physical constructions, squared paper and spreadsheets to solve your problems.

Please *read the criteria* carefully to ensure you are demonstrating all Learning Outcomes at a satisfactory or higher level. *Submit all work* done on tasks including rough notes as appendices whether or not you believe you have completely solved a particular problem. To met the criteria at least at the satisfactory level you need to genuinely attempt (but not necessarily completely solve) all four problems.

**PROBLEM 1:** *The Integer Problem*: *Given any 9 integers, show that you can choose two of them that have a difference that is a multiple of eight.*

*Parts A and B can be attempted in collaboration with another student*. (Keep a record of your collaboration or any individual contribution in a Log book.)

A: Three sets of nine random integers (from 1 to 100 inclusive) have been generated using a CAS-enabled graphing calculator (See Figure 1.)

*Figure 1.* Three sets of nine random integers.

Investigate by any means, which 2 numbers from each set have a difference that is a multiple of 8. Clearly record your calculations and all reasoning.

B: Generate your own four random sets of integers. Apply your reasoning from A

C: *Independent*ly, investigate and solve the general problem (problem 1). Communicate your mathematical argument as to why it is always possible to choose two integers from a set of nine that have a difference that is a multiple of eight.

*Note*: A Spreadsheet may be useful for this problem. If you use a spreadsheet for this, or any other problem, include a printout of the file in your work. If you use any formulas you also need to print a version of the spreadsheet showing the formulas. Upload your spreadsheet to LEO with your submission of AT1. [To show formulas in all cells, go to the Formula Tab and select Show Formulas.]

**PROBLEM 2**: *The Circus Wallabies Problem*

Last night I dreamed of an imaginary circus where wallabies did tricks. Three grey wallabies and three red wallabies performed the following trick. At the beginning of the trick, the 3 grey wallabies were on the left side of a long mat partitioned into 7 squares, and the 3 red wallabies were on the right side. Each wallaby had its own square with an empty square in the middle.

The wallabies can only move in two different ways. They can **hop** to the adjacent square *if* it is empty. If the adjacent square was not empty, they could **jump** over one other wallaby to an empty square.

The grey wallabies moved only from left to right. The red wallabies moved only from right to left. When the trick was over all the wallabies had swapped places. All the grey wallabies were on the were on the right side, and all the red wallabies were on the left side.

*Parts A and B can be attempted in collaboration with another student*. (Keep a record of your collaboration or any individual contribution in a Log book.)

- The wallabies needed 15 moves to swap places. Explain using words and diagrams how this was possible. How many moves were hops? How many moves were jumps?
- What is the smallest number of moves that 5 grey wallabies and 5 red wallabies would need to swap places (using a long mat partitioned into 11 squares). Explain using words and diagrams how this was possible. How many moves were hops? How many moves were jumps?

*Independent work.*

- What if 20 grey wallabies and 20 red wallabies were involved? How many hops would be needed? How many jumps would be needed? How many moves would be made in total? Communicate clearly the reasoning you used to solve the problem.
- Solve the problem for any number of wallabies?

**PROBLEM 3:** *The Hexagon Problem*

*Figure 2*. A tangram.

Using all pieces from one tangram set (see Figure 2) it is possible to create many different figures.

- Five
*convex hexagons*can be created. Find all five of these, discussing your approach. What is the same about the figures created and what is different? - Finding
*concave hexagons*. Develop as systematic approach – and communicate this clearly using words and diagram – to find as many concave hexagons as you can.

*Note:* The emphasis is on communicating your system and thus demonstrating mathematical thinking. A file with multiple tangram sets is available in the resource folder on LEO

**PROBLEM 4**: *The Ramp Problem- A real-world modelling problem*

Source: http://accessadvocates.com/wheelchair-ramps-important/ Source: https://www.networx.com/article/wheelchair-ramp-cost

Ramps are important. If you use a wheelchair for mobility, getting to places can be difficult. Just because you are in a wheelchair should not mean you are denied access to any building (e.g., Disability Discrimination Act (Access to Premises – Building) 2010). Hence, wheelchair ramps are critical. In addition to the unavailability of disabled access many ramps pose many problems to their users. Issues to be considered in ramp design according to the National Construction Code (NCC) (2019) based on the relevant **Australian Standard** include:

*Maximum transition gradient*: Transition from one surface to another can pose issues for wheelchairs. Ramps must have a gradient no more than 1 : 8.*Rolling resistance*which effects traversability and stability.*Width*impacting manoeuvrability and turning*Camber**Ramp length vs Gradient*(shorter ramps can be steeper, accounting for fatigue of users, but the maximum ramp length (of 9 metres) must have a gradient of 1:14 (NCC, 2019).

The controlling dimensions for **ramps**, handrails and kerbs are also detailed in the relevant **Australian Standard**. A ramp at a gradient of 1:14 should have a landing at 9 metre intervals.

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*Ramps: Are they legal and or fit for purpose?*

Identify 3 ramps – specify the specific location (they must be at different buildings/ locations). Include at least one ‘old’ ramp if possible. Include a photograph of each ramp. Ensuring you respect the owners and users of the ramp, make appropriate measurements. Sketch or draw a scale diagram showing all important measurements of your selected ramps.

The Measure App on your iPhone or iPad, or similar app on an android will be useful in making measurements. |

Evaluate your selected ramps to answer the two questions. Firstly, do they meet the Australian Standards? If not, what changes are required. Secondly, consider how *fit for purpose* your three ramps are. Propose some criteria (a mathematical model) and use these to order your three ramps from best to least *fit for purpose*. Make arguments for proposed change to improve one or more of your ramps.

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