Stage 2 Mathematical Applications

STAGE 2 MATHEMATICAL APPLICATIONS

ASSESSMENT TYPE 1: SKILLS AND APPLICATIONS TASK 1

Purpose To demonstrate your ability to:  accurately apply the mathematical concepts and relationships that you have learned in class to solve a range of matrices questions set in different contexts  effectively and appropriately communicate relevant information within your solutions.

Description of assessment This assessment allows you to show your skills in understanding and appropriate use of the mathematical concepts and relationships in the following:

(a) Subtopic 4.1: Using Matrices to Organise Information – Costing and Stock Management (b) Subtopic 4.2: Application of Matrices to Network Problems.

Assessment conditions This is a supervised assessment. Provide complete working for all calculations. Use electronic technology where appropriate.

Learning Requirements Assessment Design Criteria Capabilities

1. Understand fundamental Mathematical Knowledge and Skills and Their Communication mathematical concepts Application Citizenship and relationships. The specific features are as follows: 2. Identify, collect, and Personal . MKSA1 Knowledge of content and understanding of Development organise mathematical mathematical concepts and relationships. information relevant to Work investigating and finding . MKSA2 Use of mathematical algorithms and solutions to techniques (implemented electronically where Learning questions/problems taken appropriate) to find solutions to routine and complex from social, scientific, questions. economic, or historical . MKSA3 Application of knowledge and skills to answer contexts. questions in applied contexts. 3. Recognise and apply the Mathematical Modelling and Problem-solving mathematical techniques needed when analysing The specific features are as follows: and finding a solution to a . MMP1 Application of mathematical models. question/problem in . MMP2 Development of mathematical results for context. problems set in applied contexts. 4. Make informed use of . MMP3 Interpretation of the mathematical results in the electronic technology to context of the problem. provide numerical results and graphical . MMP4 Understanding of the reasonableness and representations. possible limitations of the interpreted results, and recognition of assumptions made. 5. Interpret results, draw Communication of Mathematical Information conclusions, and reflect on the reasonableness of The specific features are as follows: these in the context of the . CMI1 Communication of mathematical ideas and question/problem. reasoning to develop logical arguments. 6. Communicate . CMI2 Use of appropriate mathematical notation, mathematical ideas and representations, and terminology. reasoning using appropriate language and representations. 7. Work both independently and cooperatively in planning, organising, and carrying out mathematical activities.

of 10 Stage 2 Mathematical Applications task for use in 2011 08f720c98eafe3567e7fba19f1447290.doc (revised January 2013) © SACE Board of South Australia 2010 PERFORMANCE STANDARDS FOR STAGE 2 MATHEMATICAL APPLICATIONS

Mathematical Knowledge and Mathematical Modelling and Problem- Communication of Skills and Their Application solving Mathematical Information

A Comprehensive knowledge of content Development and effective application of Highly effective communication and understanding of concepts and mathematical models. of mathematical ideas and relationships. Complete, concise, and accurate solutions to reasoning to develop logical Appropriate selection and use of mathematical problems set in applied contexts. arguments. mathematical algorithms and techniques Concise interpretation of the mathematical results in Proficient and accurate use of (implemented electronically where the context of the problem. appropriate notation, appropriate) to find efficient solutions to representations, and complex questions. In-depth understanding of the reasonableness and terminology. possible limitations of the interpreted results, and Highly effective and accurate application recognition of assumptions made. of knowledge and skills to answer questions set in applied contexts.

B Some depth of knowledge of content and Attempted development and appropriate application Effective communication of understanding of concepts and of mathematical models. mathematical ideas and relationships. Mostly accurate and complete solutions to reasoning to develop mostly Use of mathematical algorithms and mathematical problems set in applied contexts. logical arguments. techniques (implemented electronically Complete interpretation of the mathematical results in Mostly accurate use of where appropriate) to find some correct the context of the problem. appropriate notation, solutions to complex questions. representations, and Some depth of understanding of the reasonableness terminology. Accurate application of knowledge and and possible limitations of the interpreted results, and skills to answer questions set in applied recognition of assumptions made. contexts.

C Generally competent knowledge of Appropriate application of mathematical models. Appropriate communication of content and understanding of concepts Some accurate and generally complete solutions to mathematical ideas and and relationships. mathematical problems set in applied contexts. reasoning to develop some logical arguments. Use of mathematical algorithms and Generally appropriate interpretation of the techniques (implemented electronically mathematical results in the context of the problem. Use of generally appropriate where appropriate) to find mostly correct notation, representations, and solutions to routine questions. Some understanding of the reasonableness and terminology, with some possible limitations of the interpreted results, and inaccuracies. Generally accurate application of some recognition of assumptions made. knowledge and skills to answer questions set in applied contexts.

D Basic knowledge of content and some Application of a mathematical model, with partial Some appropriate understanding of concepts and effectiveness. communication of relationships. Partly accurate and generally incomplete solutions to mathematical ideas and Some use of mathematical algorithms mathematical problems set in applied contexts. reasoning. and techniques (implemented Attempted interpretation of the mathematical results Some attempt to use electronically where appropriate) to find in the context of the problem. appropriate notation, some correct solutions to routine representations, and questions. Some awareness of the reasonableness and possible terminology, with occasional limitations of the interpreted results. Sometimes accurate application of accuracy. knowledge and skills to answer questions set in applied contexts.

E Limited knowledge of content. Attempted application of a basic mathematical model. Attempted communication of Attempted use of mathematical Limited accuracy in solutions to one or more emerging mathematical ideas algorithms and techniques (implemented mathematical problems set in applied contexts. and reasoning. electronically where appropriate) to find Limited attempt at interpretation of the mathematical Limited attempt to use limited correct solutions to routine results in the context of the problem. appropriate notation, questions. representations, or Limited awareness of the reasonableness and terminology, and with limited Attempted application of knowledge and possible limitations of the results. skills to answer questions set in applied accuracy. contexts, with limited effectiveness.

of 10 Stage 2 Mathematical Applications task for use in 2011 08f720c98eafe3567e7fba19f1447290.doc (revised January 2013) © SACE Board of South Australia 2010 STAGE 2 MATHEMATICAL APPLICATIONS SKILLS AND APPLICATIONS TASK 1 SOLUTIONS MATRICES

1. (a) Draw and label a network with connectivity as shown in this matrix. A B C D E A 0 1 0 0 1 B   0 0 0 0 0 C 0 1 0 1 1 D   0 0 0 0 1 E 1 1 1 0 0

Requires E B appropriate representation of a network to demonstrate communication of mathematical information. D C

(2 marks)

(b) One of the locations seems to be unusual. Which one is it and why? If the nodes represented locations inside a video store what could this location represent?

B does not have any arcs leaving it  B could represent the checkout counter (or exit) in a video store.

(2 marks)

of 10 Stage 2 Mathematical Applications task for use in 2011 08f720c98eafe3567e7fba19f1447290.doc (revised January 2013) © SACE Board of South Australia 2010 2. The table below shows some of the charges made for various services by telephone companies.

Company Local call Long distance Mobile call Text Message ($) call ($) ($) message ($) bank ($) W 0.29 1.00 0.58 0.15 0.20 X 0.27 0.95 0.69 0.12 0.20 Y 0.23 1.07 0.72 0.05 0.12 Z 0.26 0.87 0.83 0.10 0.15

(a) What service attracts the smallest charge? Text message (1 mark) (b) A customer in one month made 45 local calls, 15 long distance calls, 106 mobile calls and sent 345 text messages. Use matrix methods to find the cost for this customer at the various companies. Which company is the cheapest for him?  45  0.29 1.00 0.58 0.15 0.20  15  0.27 0.95 0.69 0.12 0.20   Cost = C x N =    106 0.23 1.07 0.72 0.05 0.12     345 0.26 0.87 0.83 0.10 0.15  0 

141.28 140.94 =   Cheapest company is Y charging $119.97. 119.97   Routine questions 147.23 (1- and 2-step) that require selection (3 marks) and use of mathematical (c) His brother spends most of his working day on the road so makes use of the message algorithms to find bank service. A typical month’s use would be 50 local calls, 120 long distance calls, 135 solutions. Together mobile calls, 200 text messages and 150 message bank uses. Use matrix methods to Questions 1 and 2 find which company offers him the best deal? give a first impression of Cost = C x B knowledge of content.  50  272.80 120   274.65 = C x 135      265.10 200   271.95 150 Company Y charging $265.10 is also cheapest for his brother. (2 marks)

of 10 Stage 2 Mathematical Applications task for use in 2011 08f720c98eafe3567e7fba19f1447290.doc (revised January 2013) © SACE Board of South Australia 2010 3. Scott has the job of keeping the frozen pizza stocked in the freezer at the local supermarket. The store keeps three brands of pizza, Apizza, Best and Choice in three different sizes Family, Regular and Small. At the beginning of the day Scott knew that he had 11 Family, 9 Regular and 6 Small pizzas of the Apizza brand, 8 Family, 12 Regular and 11 Small pizzas of the Best brand and 14 Family, 10 Regular and 15 Small pizzas of the Choice brand. (a) Represent this information as matrix B . Moving from information F R S being provided in an unstructured format to a A B = 11 9 6  structured format (e.g. B  8 12 11 matrix form) is   considered a complex C 14 10 15 process. (2 marks) Matrix E shows the stock at the end of the day. F R S E = A 6 8 5 B   4 6 5 C 7 4 9

(b) Calculate N  B  E and explain what information N contains.

F R S N = B – E = A 5 1 1 B   4 6 6 C 7 6 6

Matrix N shows how many of each brand and size pizzas are sold on the day. (2 marks) (c) The pizzas are priced so that all brands make the same profit for each of the three pizzas. The Family pizza gives a profit of $2.35, the Regulars earn $1.70 and the Small pizzas give a profit of $1.05. Write the information as a column matrix and use it to help find the total profits made on the pizza for the day. 2.35   P  1.70 Total profit = 1 1 1 N  P 1.05 5 1 1 2.35      1 1 1 4 6 6  1.70  73.35 7 6 6 1.05  total profit = $73.35. (3 marks) (d) Show how you can use the matrices above to determine: (i) what size pizza seems to be the most popular Adding up the columns of matrix N there were 16 Family, 13 Regular and 13 Small pizzas sold  Family pizza is most popular. (ii) what brand has the largest market share on the day. Adding up the rows of matrix N Choice brand has the largest market share with 19 pizzas sold. (2 marks)

of 10 Stage 2 Mathematical Applications task for use in 2011 08f720c98eafe3567e7fba19f1447290.doc (revised January 2013) © SACE Board of South Australia 2010 4. The diagram represents the lines of communication between 5 sections in a manufacturing facility. (a) Draw up a connectivity matrix R to describe the network below.

 0 1 0 1 0 B    0 0 1 0 0 R =  0 1 0 0 1   A  0 0 1 0 1 E C 01 1 0 0 0

(2 marks) (b) Calculate R 2 and explain what information it contains.

0 0 2 0 1   0 1 0 0 1 R 2  1 1 1 0 0   1 2 0 0 1 0 1 1 1 0

R 2 contains the number of 2-stage communication lines between the sections, i.e. communication between two sections via another, e.g. A talks to B then talks to C. Provides an opportunity to interpret (2 marks) mathematical results in the context of the (c) What is the minimum number of stages it takes for B to communicate with A? Explain. question. There are similar opportunities in each question that B  C  E  A  3 stages are needed for B to communicate with A. provide evidence of application of knowledge and skills to answer questions (1 mark) in applied contexts. (d) Calculate the matrix Q  R  R 2  R3.

1 4 2 1 3   1 2 2 0 1 Q  1 4 2 1 2   2 4 3 1 2 1 3 3 1 2

of 10 Stage 2 Mathematical Applications task for use in 2011 08f720c98eafe3567e7fba19f1447290.doc (revised January 2013) © SACE Board of South Australia 2010 (i) What is the meaning of Q4,2  4?

Q4,2  4 means that there are 4 ways that D can communicate with B in three or less stages.

(ii) One division still does not communicate with another in either 1, 2 or 3 stage connections. Which is it?

B to D as Q2,4  0

(3 marks) (e) What single step alteration or addition do you think would improve the system? Why?

(Answers will vary). Give B direct communication with D. This means that all sections can communicate with each other in 3 or less stages.

(2 marks) (f) Explain the limitations of using the matrix model in this situation.

The matrix model indicates how many but not what the communication links are between Part f) is one of the sections so some communication links between sections pass back and forth, e.g. several Q  2 as B  C  E  B and B  C  B. opportunities to 2,2 discuss the limitations of the Need to be careful if you also considered 4 stage links, e.g. B  C  B  C would interpreted results. describe a link from B to C but it has already been counted as a 2 stage link. Each response Our matrix model doesn’t give the number of unique links. The model assumes that direct must relate to the and indirect links are equally important as no higher weighting has been given to direct context of the question. links.

(2 marks)

of 10 Stage 2 Mathematical Applications task for use in 2011 08f720c98eafe3567e7fba19f1447290.doc (revised January 2013) © SACE Board of South Australia 2010 5. An enterprising farmer decides to set up a small factory to supply the local vineyards with her innovative machines for use in grape production. She makes a small cultivator for ploughing between rows of vines, a mower for keeping down the grass and a spray unit for spraying the vines for downy mildew. She developed the following matrix M, which shows the number of units of different materials needed to make each design.

Steel Paint Wheels Labour M = Cultivator 7 2 4 15 Mower   5 3 3 18 Sprayer 4 3 2 14

She also made a cost matrix C of the unit cost in dollars.

38  9  C    85   42 Her initial orders look very promising. These original orders are shown in the matrix O. Question 5 provides an opportunity to O  11 19 23 demonstrate highly effective and accurate application of matrix knowledge Use matrix methods to find: and skills to find (a) The quantities needed to fill the initial orders. solutions to questions set in applied contexts. 7 2 4 15   Q  O  M  11 19 23 5 3 3 18  264 148 147 829 4 3 2 14

Parts a) and b) do (2 marks) not provide the formulae and (b) The cost of each machine. therefore require use of mathematical Each machine cost: algorithms and techniques 38 7 2 4 15 1254 (implemented   electronically) to    9    E  M  C  5 3 3 18   1228 find solutions to 85 complex questions. 4 3 2 14    937  42 Cost of production of: cultivator $1254, mower $1228, sprayer $937.

(2 marks)

of 10 Stage 2 Mathematical Applications task for use in 2011 08f720c98eafe3567e7fba19f1447290.doc (revised January 2013) © SACE Board of South Australia 2010 (c) The total cost of materials needed to fill the initial orders. Total Cost, T = O x E 1254    11 19 23 1228  58677  937  Total Cost = $58677 (2 marks) (d) The retail price of each machine if she aims for a 60% profit on each item. Price P = 1.6 x E 1254 2006.40      1.6 1228  1964.80  937  1499.20 Retail prices: cultivator, $2006.40; mower $1964.80; sprayer $1499.20. (2 marks) (e) Calculate the total income from the initial orders. Total Income, I = O x P 2006.40    11 19 23 1964.80  93883.20 1499.20 Total Income = $93883.20 (2 marks) (f) Use your answers to calculate the total profit made on the initial orders. Total Profit = Total Income – Total Cost = I – T = 93883.20 – 58677 Total Profit = $35206.20

(2 marks) (g) How reasonable is your answer. Discuss the limitations of using matrix models.

Reasonableness: Model doesn’t take into account all costs, e.g. overhead costs, transport, wastage, so that the actual profit may be different. It is a useful/efficient method when all the machines have the same raw materials. It lets us organise the data efficiently but care needs to be taken to multiply matrices in the correct order to get meaningful results. In some parts of the question it would have been just as easy to calculate results without the use of matrix methods.

(2 marks)

of 10 Stage 2 Mathematical Applications task for use in 2011 08f720c98eafe3567e7fba19f1447290.doc (revised January 2013) © SACE Board of South Australia 2010 6 Four children are observed in a play situation and an impression of any dominance relations is made. This has been represented on the network diagram. (a) Represent this information in the form of a matrix D. A 0 1 0 1 1 0 1 1 D    1 0 0 1 B   D 1 0 0 0

(2 marks) (b) Calculate D 2 . 2 0 1 1 2 1 0 2 D 2    1 1 0 1   0 1 0 1 (2 marks)

2 (c) Describe what is meant by D2,4 and verify it from the diagram.

2 D2,4  2 indicates that B has 2 second order dominances over D as B  A  D and B  C  D (2 marks)

(d) Use the model S  D  1 D 2 to decide the ranking of leadership among the children. 2 Parts a), b) and d) provide the 1 1  1 1 2 1 2   4  mathematical   2 1 1 2 5 1  results that are S   2  V   2  used to inform the  1 1 1   1  development of 1 2 2 0 1 2 3 2  1 1    logical arguments  1 2 0 2   2  about dominance relationships Ranking from most dominant to least dominant is B, A, C, D. through highly effective (3 marks) communication of 1 2 mathematical ideas (e) Why is 2 D used in the supremacy matrix S. and reasoning.

1 2 2 D is used to show that first order dominance is twice as significant as second order dominance when ranking for leadership. (1 mark) (f) Briefly discuss any limitations of this mathematical model.

The model uses an impression of dominance which may not actually exist in reality. Second stage dominance includes being given power over itself from A  B  A and A  C  A . If another child entered the group the group dynamics may change. Also group dynamics likely to change with classroom environment. (2 marks)

of 10 Stage 2 Mathematical Applications task for use in 2011 08f720c98eafe3567e7fba19f1447290.doc (revised January 2013) © SACE Board of South Australia 2010