Further Mathematics with Ollie

Sunshine College, 2017

This table of contents is hyperlinked. Click the title you want to jump to the relevant section.

Course outline and important information 2

Progress Checks, how do they work? 4

Weeks 1-15 (Link) 5

Week 16: Introduction to Matrices 5

Week 20: Matrix dominance, transitions, and simultaneous equations. 11

Week 21: Transition matrices 19

Week 22: Intro to Networks 22

Week 23: Euler, Hamilton, Djikstra, and Prim 25

Week 24: Cuts, Activity Networks, Critical Paths, and the Hungarian Algorithm 30

Week 25: Crashing! 37

Week 32: Revision lesson 39

Course outline and important information

This document can be found in full, along with unit notes, at: http://www.ollielovell.com/further

Contact Ollie anytime at oliverlovelltas@gmail.com

Unit 3

Transition	(2016)	Core	Data Analysis	Univariate Data	Weeks 36 – 38
Semester 1	Term 1			Comparing Data Distributions	Weeks 1 – 2
				Regression	Weeks 3 – 5
				Time Series	Weeks 6 – 7
				SAC 1 – 40 Marks	Weeks 8 – 9
	Term 2		Recursion & Financial Modelling	Recurrence Relations	Weeks 10 – 11 (1,2)
				Interest & Depreciation	Weeks 12 – 14 (3,4,5)
				Loans, Investments & Asset Values	Weeks 15-16 (6,7)
				SAC 2 – 20 Marks	Week 17 (8)
				Unit 3 Revision & Practice Exam	Week 18 (9)

Unit 4

Semester 1	Term 2	Module 1	Matrices	Matrices	Weeks 19 – 21 (10,11, Term3 wk1)
Semester 2	Term 3	Module 1	Matrices	SAC 3 – 20 Marks	Week 22 (2)
		Module 2	Networks & Decision Mathematics	Undirected Graphs & Networks	Weeks 23 – 24 (3,4)
				Directed Graphs & Networks	Weeks 25 – 26 (5,6)
				Review	Week 27 (7)
				SAC 4 – 20 Marks	Week 28 (8)
				Exam Revision	Weeks 29 – 30 (9,10)
	Term 4			Exam Revision	Weeks 31 – 33 (Term 4 wk1, 2, 3)
				SWOTVAC	Week 34 (4)
				Exam Period	Weeks 34 – 38 (4,5,6,7,8)

Modules not studied: Module 3 – Geometry & Measurement, Module 4 – Graphs & Relations.

HOMEWORK AND PROGRESS CHECKS

In addition to the above, each week students will complete a short progress check based upon content from the previous three weeks. Progress checks are designed to keep track of student learning, and indicate whether students are keeping up with the content of the course. These progress checks will not contribute towards students’ final marks, but will be used to indicate whether the students are keeping up with the course content. This progress check will take place in the single period scheduled in the timetable. For justification of this distributed practice approach, see this research article.

Weekly Questions will be set, with questions drawn from previous exams. Questions for progress checks will be drawn directly from these Weekly Questions (i.e, a small selection of questions will be selected from the larger selection of Weekly Questions). Students are expected to complete all assigned homework in preparation for progress checks. Homework will be distributed in the first lesson each week, students not present during the first lesson must collect homework from the mathematics staff room.

RESOURCES

All students will need the following (tick once obtained)

A blank exercise book for rough work.
A blank exercise book to be a bound exam reference. (for further information on bound references, see here)
TI-nspire CAS CX Calculator
Standard writing implements such as pens, pencils, eraser, sharpener, and ruler

ASSESSMENT

Outcome 1 - Skills

On completion of this unit the student should be able to define and explain key concepts and apply related mathematical techniques and models as specified in the Areas of Study in routine contexts.

Outcome 2 - Applications

On completion of this unit the student should be able to select and apply the mathematical concepts, models and techniques as specified in the Areas of Study in a range of contexts of increasing complexity.

Outcome 3 – Technology (TI-nspire CAS CX Calculator)

On completion of this unit the student should be able to select and appropriately use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches.

These Outcomes will be assessed via school assessed coursework (SACs), and the external examination.

Students must satisfy all three Outcomes in each Unit in order to satisfactorily complete the Unit.

SCHOOL ASSESSED COURSEWORK (SACs): Each sac will be spread across multiple periods

Unit 3 SAC 1 – Data Analysis – 5hrs, 13% Unit 3 SAC 2 – Recursion & Financial Modelling – 3hrs, 7%

(Application Task) (Modelling & Problem-Solving Task)

Unit 4 SAC 3 – Matrices – 3hrs, 7% Unit 4 SAC 4 – Networks & Decision Mathematics – 3hrs, 7%

(Modelling & Problem-Solving Task) (Modelling & Problem-Solving Task)

School assessed coursework contributes a total of 34% to a student’s study score, with the remaining 66% from external examinations. Attendance to SACs is compulsory. Students who are absent without an accompanying doctor’s certificate will not be provided with an alternate opportunity to attempt the missed SAC. .

EXTERNAL EXAMS

Exam 1 – Multiple Choice – 1.5hrs, 33% Exam 2 – Short Answer – 1.5hrs, 33%

The two end of year exams contribute a total of 66% to a student’s study score.

Progress Checks, how do they work?

Ollie hands out progress checks, you do them under test conditions (atm you can use your log books, but that may change).

Please ensure that you keep the progress crease free so that Ollie can scan them in easily.

After the time is up (approx 2 mins per mark) everyone immediately puts pens down and removes everything from their desk except for the progress check that you just did (including pens, log books, bags, etc)
Everyone takes a red pen (Ollie has lots of them) and we go through the solutions together. Self-marking
You hand in your progress checks so Ollie can make a copy of them and enter your results into his gradebook (these marks don’t count towards your final mark but helps him track your progress.

Weeks 1-15 (Link)

The content from Weeks 1 to 20 can be found here: https://docs.google.com/document/d/1fqVI6fHq6P3bgoUVgZfQny5vQciLqujLZ_fY57VYR9I/edit?usp=sharing

(This doc was getting really big and taking too long to load!)

Week 16: Introduction to Matrices

Intended Learning Outcomes

By end of week, students will be able to…

Describe what a matrix is
Determine the order of a matrix
Transpose a matrix
Apply rules to determine the values of a matrix’s elements
Multiply matrices and interpret the results.

Learn:

What is a matrix?

What are matrices (the plural of matrix) used for?

How we’ll use them: To concisely represent and manipulate information. Especially data tables (as above) or networks.
How you already use them: Computer graphics make use of matrices to represent 3D objects on 2D surfaces.
How you might use them in future: Chemistry makes use of matrices in various ways, particularly since the use of quantum theory to discuss molecular bonding and spectroscopy. Examples are the overlap matrix and the Fock matrix used in solving the Roothaan equations to obtain the molecular orbitals of the Hartree–Fock method. (What the Fock?)

The Order of a Matrix

The order of a matrix is given by number of rows x (pronounced ‘by’) number of columns

I do: What is the order of the matrix, D, displayed above.
You do: What is the order of the following matrix?
Question: Would the following be a ‘row’ or a ‘column’ matrix?

Transpose of a matrix

Transpose occurs when you switch the rows and columns of a matrix.

I do: What is the transpose of the matrix pictured above?

Question: What happens to the order of a matrix when you transpose it?

You do: What is the transpose of the following matrix

(Solution to be worked out in following section)

Other transpose questions
2016, Exam 1, Question 1

Matrices on CAS.

Enter a matrix: Divide button, select → Enter values (using tab) → ‘ctrl’ + ‘var’ → select a letter
Transpose a matrix: type the matrix’s letter → menu → matrix & vector → transpose.

I do: Use this to check my answer from above.
You do: Use this to check your answer from above.

Rules for matrix elements

I do: 2014, Exam 1, Question 6
You do: 2016, Exam 1, Question 5 (hint, make the matrix!)

Interpreting a matrix (simple)

I do: 2011, Exam 1, Question 1
You do: 2016, Sample 1, Question 1

Interpreting a matrix (a touch harder)

I do: 2012, Exam 1, Question 7
You do: 2014, Exam 2, Question 1
Also see: 2013, Exam 1, Question 5

Multiplying Matrices (manually) (If you want to understand what is happening with these matrix multiplications you can check out this video).

Starting points for matrix multiplication

Defined and Undefined expressions, and results

‘Defined’ just means ‘you can do’.

You do: 2011, Exam 1, Question 4
You do: 2013, Exam 1, Question 2
Also: 2012, Exam 1, Question 6 (harder!)

Matrix Multiplication

I do:

You do:

2016, Exam 1, Question 2 (medium)
2013, Exam 1, Question 7 (harder)
Question: Which numbers must be the same if we’re multiplying an (a x b) with a (c x d) matrix?

Combining Multiplication and Interpretation

2012, Exam 1, Question 6
2011, Exam 2, Question 2
2016, Exam 2, Question 1
2016, Exam 1, Question 4

Questions that you can (and should) do on your CAS

2011, Exam 1, Question 2
2012, Exam 1, Question 1
2013, Exam 1, Question 1
2014, Exam 1, Question 1

Week 20: Matrix dominance, transitions, and simultaneous equations.

Intended Learning Outcomes

By end of week, students will be able to…

Determine Dominance
Apply transition matrices
Calculate a Matrix’s Determinant
Find the inverse of a Matrix
Use matrices to solve simultaneous equations

Learn:

One-step Dominance

I do: Create a one-step dominance matrix for the following situation:

Answer

You do: Create a 1 step dominance matrix for the network in Checkpoints 2016, Question 83.

Two-step dominance

Generated by Taking the square of a one-step dominance matrix.

I do: Create a 2 step dominance matrix for the network above (do on CAS)

Answer

You do: Create a 2 step dominance matrix for the network in Checkpoints 2016, Question 83

Total Dominance

Total dominance calculated by Total=D + D2 then summing along rows.

I do: Calculate the dominant player from the network above

Answer

You do: Checkpoints, 2016, Question 83 and 84.
You do: Checkpoints, 2013, Module 5, Question 5 (for some reason there are two Qs called 2013 Q5, thisi s the one that also has ‘Module 5’ in front of it : )

Transition Matrices

Transition Matrices describe how transitions are made between two states.

I do: 2007, Exam 1, Question 8.
You do: 2009, Exam 1, Question 5
You do: 2014, Exam 1, Question 8
You do: Checkpoints 2016, Question 126.

The identity matrix

Identity matrices have 1s along the main diagonal with zeroes in all other locations.
For the identity matrix, A x I = I x A = A for any matrix ‘A’ that is a square (n x n) matrix

The Inverse Matrix, A-1.

Two matrices are inverse if, when multiplied, the answer is the identity matrix. A x A-1=I

You do: Find out whether A and B are inverse matrices if

Condition for an inverse matrix to exist

A matrix has an inverse if its determinant is non-zero.

Determinant of a 2 x 2 matrix.

You do: Find the determinant of
You do (if bored): Find the determinant of
You do: Have a look on your CAS and see if you can work out how to check your answer using your CAS.

Finding the inverse of a 2 x 2 matrix

I do (on CAS): Find the inverse of
You do (on CAS): Find the inverse of
You do: 2011, Exam 1, Question 8.

Setting up simultaneous equations from a matrix multiplication

I do: Write the following matrix multiplication as a set of simultaneous equations:
You do: Write the following matrix multiplication as a set of simultaneous equations:

Setting up a matrix multiplication from simultaneous equations

I do: Represent the following as a matrix multiplication:
You do: Represent the following as a matrix multiplication

Solving Simultaneous Equations with Matrices

The following is explained using matrix algebra. You don’t need to know it. It’s just included here for completeness. What you do need to know is the hollow dot point below.

If AX=C then X is found by calculating A-1C.

I do: Solve the following simultaneous equations by matrix multiplication:
Solution:

OR (easier on CAS)

You do: Solve the following simultaneous equations by matrix multiplication
Harder: 2013, Exam 1, Question 6.

Simultaneous equations: Does a solution exist?

For A X = C to have a solution, det(A) must be non-zero.

You do: 2013 Exam 1, Question 4.

More considerations for defined expressions

You do: Do the following on your CAS:
Solution:

If A+B=C then C11=A11+B11. Thus…

for A+B to be defined, A and B must be of the same order (The same is true for subtraction).

Is the following statement true: ‘For A2 to exist, A must be a square matrix’ ?

For A2 to exist, A must be a square matrix.

You do (later,it’s pretty hard): 2013, Exam 1, Question 9

Permutation Matrices (We’ve already seen these, just haven’t named them)

A permutation matrix is a matrix with only 1s and 0s (thus, it’s a ‘binary’ matrix) with only one 1 in each row and column.
A permutation matrix is used to rearrange the elements in another matrix.

Questions to Focus on for PC

2012, Exam 1, Question 1, 2, 7
2014, Exam 1, Q 1, 6
2016, E1, Q1

Week 21: Transition matrices

Intended Learning Outcomes

By end of week, students will be able to…

Construct transition matrices from transition diagrams and vice versa
Use transition matrices to track change from one to multiple periods.
Conceptualise matrix transitions as recurrent relations

Learn:

Constructing a transition matrix from a transition diagram.

I construct a transition matrix for the diagram in 2011, Exam 1, Question 5
You do: 2012, Exam 1, Question 5

Applying a transition diagram to interpret a transition

I do: (Relating to the diagram from 2011, Exam 1, Question 5.) How many people would likely vote for Rob 1 week after the election campaign begins?
You do: (Relating to the diagram from 2012, Exam 1, Question 5), If in one week 250 families ate at Big Burgers and 300 ate at Fast Fries, how many would eat at fast fries the following week?

Applying a transition matrix to interpret change after one transition

I do: (Relating to the diagram from 2011, Exam 1, Question 5.) How many people would likely vote for Rob 1 week after the election campaign begins? ← Except using matrix multiplication.
You do: 2014, Exam 1, Question 4
You do: 2012, Exam 2, Question 3, part a

Understanding transition matrices as recurrent relations (And results after multiple periods with a formula)

If the initial state matrix is given by S0, the state matrix after n transitions is given by Sn=TnS0.

I do: 2011, Exam 1, Question 6
You do: If the families continue to eat at the restaurants (from 2012, E1, Q5, 250 families ate at Big Burgers and 300 ate at Fast Fries) in the same pattern, which restaurant will be more popular after 5 weeks and how many more families will be visiting this more popular restaurant?

‘In the long term’: Steady state solutions to Transition matrices

You do:

Question: How many customers will drink Brazilian coffee after 5, 10, 20, 40, and 100 days (following the day described above)?
(Ollie to draw a quick sketch of what this looks like on the board).
Also see: 2014, Exm 1, Question 3

Results after multiple periods (using brute force)

1. Enter matrix multiplication (TS0)for first transition
2. Copy paste original T to next line multiply by answer, ctrl (-)
3. Hit and keep on hitting enter!

I do: 2012, Exam 2, Question 3, part c iii.
You do: 2013, Exam 2, question 2, part ci.

Transition matrix modelling when the total number of units changes.

Our equation Sn=TnS0. (and the related recurrent relation Sn+1=TSn) only holds if the total number of units (customers, cars, etc) remains constant. If total number of units changes over time, we must use Sn+1=TSn + D

If no. of units changes over time (+ or - D) we must calculate Sn using a recurrent relation.

A rental starts with 90 cars, 50 located at Bendigo and 40 located at Colac. Each week, cars are returned to each location as represented in the transition matrix below.

In addition, each week 4 additional cars are added to the fleet. 2 in Bendigo, and 2 in Colac. How many cars will be in Bendigo, and how many will be in Colac after ___ weeks?

Solution:

After the first week there will be 46 cars in Bendigo and 48 cars in 43.6) cars in Bendigo and 54 (54.4) cars in Colac.

You do: 2013, Exam 1, Question 8
Also see: 2013, Exam 2, Question 2 eii.

Working backwards in matrix multiplications

Must use the relationship, if S2=TS1, then S1=T-1S2

You do: 2014, Exam 1, Question 7

Week 22: Intro to Networks

Intended Learning Outcomes

By end of week, students will be able to...

Learn:

Why networks? (intro video)

This stuff is Facebook’s bread and butter, and the maths that we’re doing underlies everything from quantitative approaches to advertising, to twitter, to political campaigning.

Graphs (in the terms of networks)

This is a graph. It is made up of vertices, edges, and faces.
Which do you think is which???

Follow up Q: How many faces are there?

Planar and non-planar graphs

A planar graph can be drawn in a way that none of the edges cross.

I do: Re-draw this graph to show that it is planar.

You do: Is the following graph planar? If so, re-draw it do demonstrate this property.

Solution

Complete Graphs

In a complete graph, each vertex is directly connected to every other vertex by an edge.

You do: 2013, Exam 1, Question 2

The degrees of vertices

The degree of a vertex is given by how many edges attach to it.

I do: 2011, Exam 1, Question 1
You do: 2012, Exam 1, Question 1
You do: 2011, Exam 1, Question 3

Drawing a graph from an adjacency matrix.

I do: Construct a graph from the following adjacency matrix

Solution

You do: 2012, Exam 1, Question 4

Euler’s Rule

For any planar graph: (vertices, edges, and faces)

You do: A connected planar graph has six vertices and nine edges. How many faces does the graph have?
Draw a connected planar graph with six vertices and nine edges.

Matching vertex degrees and edges (Working out which graph fits the diagram…)

I do: 2011, Exam 1, Question 4
You do: 2016, Exam 1, Question 5
You do: 2013, Exam 1, Question 6

Minimum edges for a connected graph:

You do: 2016, Exam 1, Question 3

Minimum number of edges for a graph to remain connected: e=v-1

Week 23: Euler, Hamilton, Djikstra, and Prim

Intended Learning Outcomes

By end of week, students will be able to...

Learn:

Eulerian trails and circuits

Eulerian trails traverse every edge of a graph (without repeats) and have exactly two vertices of odd degree (at the path’s start and end).

You do: Which of the following contain an Eulerian trail? (Name the trail)

You do: 2011, Exam 1, Question 9

Eulerian circuits traverse every edge of a graph (without repeats), begin and end at the same vertex, and require that all vertices are of even degree.

You do: Which of the following contain an Eulerian trail? (Name the trail)

You do: 2012, Exam 1, Question 5
You do: 2014, Exam 1, Question 6

Hamiltonian paths and cycles

Hamiltonian paths visit every vertex of a graph once
Hamiltonian cycles visit every vertex of a graph once except for the starting vertex, to which you return.

Note: You can only work out whether a graph contains hamiltonian paths or cycles by inspection.
You do: List a hamiltonian cycle for each of the following

You do: 2012, Exam 1, Question 2
You do later: 2012, Exam 1, Question 6

Trees

Trees have no loops, multiple edges, or cycles

You do: 2013, Exam 1, Question 1

Weighted Graphs

Weighted graphs in which the weights are physical quantities, for example distance, time or cost, are called networks.

E.g.,

Shortest path problems

When considering a weighted graph (network) we may like to know how to traverse a path with minimum distance, time, or cost. Such a situation is a ‘shortest path problem’.
(If you wanna know whaddup with Dijkstra’s path there’s a good vid on it here)

I do: Find the shortest path from A to F using Dijkstra’s algorithm.

We do: Find the shortest path from S to T using Dijkstra’s algorithm.

You do: 2013, Exam 1, Question 3

You do: 2016, Sample 1, Question 3

You do: 2014, Exam 1, Question 3

More practice with Dijkstra’s algorithm if you’re keen.

Find the shortest path from S to F

Minimum spanning trees (Prim’s algorithm)

Start at any vertex and move to the closest vertex. From there, move to the next closest vertex (from any point on your tree). Continue until all vertices are visited (BUT, don’t re-connect to your tree!)

I do: 2014, Exam 1, Question 5
You do: 2013, Exam 1, Question 3
You do: 2011, Exam 2, Question 2
You do, 2016, Exam 1, Question 4

Week 24: Cuts, Activity Networks, Critical Paths, and the Hungarian Algorithm

Intended Learning Outcomes

By end of week, students will be able to…

Determine the max flow of a flow diagram
Construct activity networks from precedence tables, and vice versa
Determine critical paths.

Learn:

Cuts

A cut separates all sinks from sources (Only add forward flowing edges)
Minimum cut, maximum flow.

I do: Calculate maximum flow of the following diagram.

You do: 2016, Exam 1, Question 2.

Activity Networks

… This is an activity network
Note: We don’t label the vertices of an activity, we label the edges. Each edge represents a task.

Precedence Tables

Precedence means ‘coming before’.
I do: Construct an activity network from the following precedence table.

Solution

You do: Construct an activity network from the following precedence table.

Dummy Activities

Sometimes dummy activities are used when two activities have the same predecessor.

I do: Construct activity network from the following precedence table.

Solution:

You do: Construct activity network from the following precedence table.

Solution:

Earliest Starting Times

EST is calculated by working out the longest path from the start of an activity network to an activity (forward scanning)

I do: 2012, Exam 2, Determine EST of activity I.
You do: 2013, Exam 2, Determine EST of activity J.

Minimum time to finish

Minimum time to complete all activities is given by the longest path from an activity network’s start to finish.

I do: 2012, Exam 2, Find the minimum time to complete all activities.
You do: 2013, Exam 2, Find the minimum time to complete all activities.

Latest Starting Times

LST is calculated by working out the smallest number possible to the start of an activity when back scanning (subtracting from minimum time to finish).

I do: 2012, Exam 2, Calculate LST for activity G
You do: 2013, Exam 2, Calculate LST for activity J

Float Time

Float time is just the difference between EST and LST for any given activity

Float time Explained. (Podcast and cleaning example)

Critical Path

Critical Path is is the path along activities with a float time of 0 (there can be more than one CP).

I do: 2012, Exam 2, part c.
You do: 2013, Exam 2, part ci.

For working

Bipartite Graphs

Hungarian algorithm (Minimising Cost)

See doc!!!

I do and you do (at same time): Based on the following cost matrices, who should do which task to minimise cost? What will the final cost be? (Note, these matrices are tabulated versions of complete weighted bipartite graphs… Ollie to explain)