2016
mathematical studies
topic 0
Transition
lecture
name:………………….………….………..
Important Symbols, Terms, Phrases & Concepts
Differential Calculus
Change
0
0
Limit
Chord (Secant)
Slope of Chord
Average Rate of Change
Tangent
Slope of Tangent
Instantaneous Rate of Change
Conjecture
Proof
Derivative
Transition Lecture
1
Introduction to Mathematical Studies
Mathematical Studies is:
 ____________________________________ ( ___________________________!!)
 _______________________________
Success is in your control!
Keys to Success
 ___________________________
 _____________________________
 ___________________________
 _____________________________
 __________________________________________________________________
Text Books
Mathematics for Year 12: Mathematical Studies 2nd Edition
Year 12 Mathematical Studies
Haese et. al.
Study and Revision Guide (2015 or 2016)
MASA
(orders for the 2016 Guide will be taken in an early lecture next year)
Graphics Calculator
You will need a graphics calculator, which must be on the list of SACE Board approved calculators.
This list can be found on the SACE website: www.sace.sa.edu.au
From the home page follow the links:
Learning, Subject List, Stage 2 Mathematical Studies, Approved Calculators
Suggested Stationery
96 page A4 5mm Graph Book(s)
(Note: All tests and exams are done on 5mm Graph Paper, it is a
good idea to get used to working on this style paper.)
A4 5mm Graph Test Pad
Let’s see how a Maths Studies Lecture is structured!!
Transition Lecture
2
An Introduction to Calculus
The development of calculus started with two questions:
1.
How do you find the _______ of shapes with __________________________? , and
2.
How do you study __________ at an ________________________ ?
It took almost ____________ for these questions to be answered formally.
Studying Change at an Instant of Time
Differential Calculus is the mathematics of ___________ and in a _________________________
world is one of the most widely used branches of mathematics.
But the fact that caused so much concern was that _________________________________ which
takes place ___________________________ gives us the expression
, which is
________________________.
The development of calculus required the idea of a ____________.
Consider the slope of the ________________ and the slope of the _________________________.
We can see that as ____________________________ ,
the
approaches the
We could write,
Slope of Tangent 
The Slope of a Chord tells us ________________________________________________________.
The Slope of a Tangent tells us _______________________________________________________.
Transition Lecture
3
Example
Consider the curve f ( x)  x x  x , where x  0 .
Let A x, f ( x) be any point on the graph of y  f (x) and let B be the fixed point 3, f (3) .
The graph of y  f (x) and the chord AB are shown below:
a)
The chord AB could be used as an initial estimate of the slope of the tangent to the curve y  f (x)
at the point B.
Draw a second chord that would be a better approximation to the slope of the tangent to the curve
y  f (x) at the point B.
b) Complete the following table, giving the slopes in the fifth column accurately to two decimal
places.
Point A
x-coordinate y-coordinate
Point B
x-coordinate y-coordinate
2.5
3
2.9
3
2.99
3
Transition Lecture
Slope of chord AB
4
c)
Use your table to make a conjecture about the slope of the tangent at point B.
In general, consider the curve _________.
Using limits to find the slope of the _______________ at _________ gives
This result could be used to prove our conjecture.
Transition Lecture
5
Notes
Transition Lecture
6
SACE Stage Two — Mathematical Studies
Program and Assessment Plan 2016
Texts: Mathematics for Year 12: Mathematical Studies
Study and Revision Guide
Term 1
2nd Edition
2015 or 2016
Working with Functions and Function Models
Haese et. al.
MASA
Chapter 1 & 5
The idea of a mathematical model and thus a functional model—Summary of the
functional models of the course and their features—Properties of a good functional
model—Working with the lower order polynomials—Graphing functional models—
Simultaneous solutions of two functional models or a functional model and a relational
model
Matrix Arithmetic and Algebra
Chapter 10
Definition of a matrix—Matrix arithmetic—Matrix algebra—The inverse and determinants
of 2  2 and 3  3 matrices—Applications.
Introductory Calculus
Chapters 2 & 3
Areas with curved boundaries—Definite integrals—Average rate of change—
Instantaneous rate of change—Idea of a limit—Derivatives at particular values and the
derivative function, from first principles—Integral of a rate
Term 1/2
Differential Calculus 1
Chapter 3, 4 & 5
Simple rules of differentiation—Composite functions and the chain rule—Product and
quotient rules—Implicit differentiation—Tangents and normals—The second derivative—
The derivative of the exponential function—The natural logarithm function—Derivatives
of logarithmic functions—Motion in one-dimension
Term 2
Differential Calculus 2
Chapter 4 & 5
Properties of curves—Rational functions—Optimisation—Exponential growth and decay
models—Surge models—Logistic growth models—Other models
Transition Lecture
7
Term 2/3
Integral Calculus
Chapter 6
Anti-differentiation—Properties of indefinite integrals—Integration rules—Review of the
definite integral—The area function—The fundamental theorem of calculus—Distance
from non-constant velocity—Definite integrals—Finding areas—Integrals of rates—
Simple differential equations
Term 3
Probability Distributions
Chapter 7 & 8
Probability distributions—Discrete & continuous random variables—The binomial
distribution—The Normal distribution.
Inferential Statistics
Chapter 7 & 8
Sampling distributions—The central limit theorem—Confidence intervals for a population
mean & a population proportion—Hypothesis testing of a population mean & a population
proportion, using the two-tailed Z-test.
Systems of Equations
Chapter 9
Verifying solutions—Solving 2  2 systems of equations— 3  3 systems with unique
solutions— 2  3 and other 3  3 systems—Parametric solutions—Further applications—
larger systems
Revision
Term 4
Revision
External 3-hour Examination
Transition Lecture
8
Year 12 Mathematical Studies
Assessment Outline
The SACE Board requires each student to be allocated a score for Skills and Applications Tasks
(i.e. Tests) and a Portfolio ( i.e. Investigations). Students also sit an External 3-hour
Examination.
Tests
Testing will be done on a topic basis and, in general, each test will carry an equal weighting. As the
year progresses, tests may contain a question from an earlier topic to give students other
opportunities to demonstrate their learning and to assist with revision.
Test
Functions used a Mathematical Models
Matrix Arithmetic and Algebra
Introductory Calculus
Differential Calculus 1
Differential Calculus 2
Mid-Year Exam
Integral Calculus
Probability Distributions
Inferential Statistics
Systems of Equations
Term
1
1
1
2
2
3
3
3
3
4
Test Conditions
Testing will usually be done during the tutorials under conditions that model the final examination.
All tests will be prepared in booklet form with graph space provided for your answers. Appropriate
steps of logic and correct answers are required for full marks.
In each test you are allowed to have a single-sided A4 page of notes. These notes are to be an
original copy, in your own handwriting. You are not allowed to bring in any other paper.
You are also allowed to have two calculators from the approved SACE Board list.
At the end of the test you will be requested to submit the question/answer booklet, your page of notes
and any scribble paper used during the test. No material pertaining to the test is to leave the room
and you are not to discuss the test with students outside your tutorial group.
A Medical Certificate is required for an absence from a test and it is the responsibility of the
student to present this to the teacher and organise an alternative time for the test. This is expected to
be done on the students next day at school.
Transition Lecture
9
Portfolio
This will be made up of 2 Investigations.
Investigations involve the concepts of mathematical investigation, formation of conjecture and proof
(or refutation). Students will also need to write introductions, methodologies, conclusions and
discussions.
Investigations may have components which are done under ‘test’ conditions, while others would be
‘take-home’. Investigations will start by being quite directed and move towards being open-ended.
Weightings
A final score will be calculated as follows:
Skills and Applications Tasks
Portfolio Tasks
External Exam
45%
25%
30%
The school assessed components, Skills and Applications Tasks and the Portfolio are subject to a
moderation process.
Andrew Bee & Lisa Lanchester
Transition Lecture
10
A suggested process to follow for the lecture/tutorial system in Maths Studies
PREVIEW
the next few pages of the
lecture booklet
LECTURE
LECTURE REVIEW
SUMMARISE THE
LECTURE
Familiarise yourself with the _______________
to be used.
Presentation of:

____________, and

_________________________________

_________ the lecture

______ about the parts that are _____________
Focus on:

_____________________ , ________________
and _______________

_____________________

_______________ , __________ and
________________________
Come and _____________ your summary.
DO THE PRACTICE
PROBLEMS
from the textbook &/or
worksheets

STAY ___________________!!

do some maths ___________________

_____ questions ____________ and __________
_____________________!
PREPARE FOR THE
TOPIC TEST

prepare your ________ of notes

learn the _____________ and ________

____________, ____________, _____________