# Specialist Mathematics

### Rationale

Mathematics is the study of order, relation and pattern. From its origins in counting and measuring it has evolved in highly sophisticated and elegant ways to become the language now used to describe much of the modern world.

### Links to Foundation to Year 10

For all content areas of Specialist Mathematics, the proficiency strands of the F–10 curriculum are still very much applicable and should be inherent in students’ learning of the subject. The strands of Understanding, Fluency, Problem solving and Reasoning are essential and mutually reinforcing.

### Representation of General capabilities

The seven general capabilities of Literacy, Numeracy, Information and Communication technology (ICT) capability, Critical and creative thinking, Personal and social capability, Ethical understanding, and Intercultural understanding are identified where they offer opportunities to add depth and richness to student learning.

### Structure of Specialist Mathematics

Specialist Mathematics is structured over four units. The topics in Unit 1 broaden students’ mathematical experience and provide different scenarios for incorporating mathematical arguments and problem solving.

## Unit 1

### Unit 1 Description

Unit 1 of Specialist Mathematics contains three topics – ‘Combinatorics’, ‘Vectors in the plane’ and ‘Geometry’ – that complement the content of Mathematical Methods. The proficiency strand, Reasoning, of the F–10 curriculum is continued explicitly in ‘Geometry’ through a discussion of developing mathematical arguments. While these ideas are illustrated through deductive Euclidean geometry in this topic, they recur throughout all of the topics in Specialist Mathematics. ‘Geometry’ also provides the opportunity to summarise and extend students’ studies in Euclidean Geometry. An understanding of this topic is of great benefit in the study of later topics in the course, including vectors and complex numbers.

‘Vectors in the plane’ provides new perspectives for working with two-dimensional space, and serves as an introduction to techniques that will be extended to three-dimensional space in Unit 3.

‘Combinatorics’ provides techniques that are useful in many areas of mathematics including probability and algebra. All these topics develop students’ ability to construct mathematical arguments.

These three topics considerably broaden students’ mathematical experience and therefore begin an awakening to the breadth and utility of the subject. They also enable students to increase their mathematical flexibility and versatility.

Access to technology to support the computational aspects of these topics is assumed.

### Unit 1 Learning Outcomes

By the end of this unit, students:

• understand the concepts and techniques in combinatorics, geometry and vectors
• apply reasoning skills and solve problems in combinatorics, geometry and vectors
• communicate their arguments and strategies when solving problems
• construct proofs in a variety of contexts including algebraic and geometric
• interpret mathematical information and ascertain the reasonableness of their solutions to problems.

### Unit 1 Content Descriptions

#### Permutations (ordered arrangements):

solve problems involving permutations  (ACMSM001)

use the multiplication principle (ACMSM002)

use factorial notation (ACMSM003)

solve problems involving permutations and restrictions with or without repeated objects (ACMSM004)

#### The inclusion-exclusion principle for the union of two sets and three sets:

determine and use the formulas for finding the number of elements in the union of two and the union of three sets.  (ACMSM005)

#### The pigeon-hole principle:

solve problems and prove results using the pigeon-hole principle. (ACMSM006)

#### Combinations (unordered selections):

solve problems involving combinations (ACMSM007)

use the notation $$\begin{pmatrix}n\\r\end{pmatrix}$$ or $${}^nC_r$$ (ACMSM008)

derive and use simple identities associated with Pascal’s triangle. (ACMSM009)

#### Representing vectors in the plane by directed line segments:

examine examples of vectors including displacement and velocity (ACMSM010)

define and use the magnitude and direction of a vector (ACMSM011)

represent a scalar multiple of a vector  (ACMSM012)

use the triangle rule to find the sum and difference of two vectors. (ACMSM013)

#### Algebra of vectors in the plane:

use ordered pair notation and column vector notation to represent a vector (ACMSM014)

define and use unit vectors and the perpendicular unit vectors i and j (ACMSM015)

express a vector in component form using the unit vectors i and j (ACMSM016)

examine and use addition and subtraction of vectors in component form (ACMSM017)

define and use multiplication by a scalar of a vector in component form (ACMSM018)

define and use scalar (dot) product  (ACMSM019)

apply the scalar product to vectors expressed in component form (ACMSM020)

examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular (ACMSM021)

define and use projections of vectors  (ACMSM022)

solve problems involving displacement, force and velocity involving the above concepts.  (ACMSM023)

#### The nature of proof:

use implication, converse, equivalence, negation, contrapositive (ACMSM024)

use the symbols for implication ($$\Rightarrow$$), equivalence ($$\Longleftrightarrow$$), and equality ($$=$$) (ACMSM026)

use the quantifiers ‘for all’ and ‘there exists’  (ACMSM027)

use examples and counter-examples.  (ACMSM028)

#### Circle properties and their proofs including the following theorems:

An angle in a semicircle is a right angle (ACMSM029)

The angle at the centre subtended by an arc of a circle is twice the angle at the circumference subtended by the same arc (ACMSM030)

Angles at the circumference of a circle subtended by the same arc are equal (ACMSM031)

The opposite angles of a cyclic quadrilateral are supplementary  (ACMSM032)

Chords of equal length subtend equal angles at the centre and conversely chords subtending equal angles at the centre of a circle have the same length (ACMSM033)

The alternate segment theorem (ACMSM034)

When two chords of a circle intersect, the product of the lengths of the intervals on one chord equals the product of the lengths of the intervals on the other chord (ACMSM035)

When a secant (meeting the circle at $$A$$ and $$B$$) and a tangent (meeting the circle at $$T$$) are drawn to a circle from an external point $$M$$, the square of the length of the tangent equals the product of the lengths to the circle on the secant. $$(AM\times BM=TM^2)$$.  (ACMSM036)

Suitable converses of some of the above results  (ACMSM037)

Solve problems finding unknown angles and lengths and prove further results using the results listed above. (ACMSM038)

#### Geometric proofs using vectors in the plane including:

The diagonals of a parallelogram meet at right angles if and only if it is a rhombus (ACMSM039)

Midpoints of the sides of a quadrilateral join to form a parallelogram (ACMSM040)

The sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of the lengths of the sides.  (ACMSM041)