Structure

Standards for the Structure dimension are introduced at Level 3.

At Level 3, students recognise that the sharing of a collection into equal-sized parts (division) frequently leaves a remainder. They investigate sequences of decimal numbers generated using multiplication or division by 10. They understand the meaning of the ‘=’ in mathematical statements and technology displays (for example, to indicate either the result of a computation or equivalence). They use number properties in combination to facilitate computations (for example, 7 + 10 + 13 = 10 + 7 + 13 = 10 + 20). They multiply using the distributive property of multiplication over addition (for example, 13 × 5 = (10 + 3) × 5 = 10 × 5 + 3 × 5). They list all possible outcomes of a simple chance event. They use lists, Venn diagrams and grids to show the possible combinations of two attributes. They recognise samples as subsets of the population under consideration (for example, pets owned by class members as a subset of pets owned by all children). They construct number sentences with missing numbers and solve them.

• The meaning of the equals sign
  
  

• Construction of Number Sentences
  
  

• Properties of Operations

At Level 4, students form and specify sets of numbers, shapes and objects according to given criteria and conditions (for example, 6, 12, 18, 24 are the even numbers less than 30 that are also multiples of three). They use venn diagrams and Karnaugh maps to test the validity of statements using the words none, some or all (for example, test the statement ‘all the multiples of 3, less than 30, are even numbers’).

Students construct and use rules for sequences based on the previous term, recursion (for example, the next term is three times the last term plus two), and by formula (for example, a term is three times its position in the sequence plus two).

Students establish equivalence relationships between mathematical expressions using properties such as the distributive property for multiplication over addition (for example, 3 × 26 = 3 × (20 + 6)).

Students identify relationships between variables and describe them with language and words (for example, how hunger varies with time of the day).

Students recognise that addition and subtraction, and multiplication and division are inverse operations. They use words and symbols to form simple equations. They solve equations by trial and error.

• Mental addition and subtraction with larger numbers
  
  
• Missing Number Sentences
  
  
• Venn diagrams
  
  
• Rules for Sequences
  
  
• Equivalence in number sentences
  
  
• Mental strategies for multiplication
  
  
• Mental strategies for division

At Level 5 students identify collections of numbers as subsets of natural numbers, integers, rational numbers and real numbers. They use Venn diagrams and tree diagrams to show the relationships of intersection, union, inclusion (subset) and complement between the sets. They list the elements of the set of all subsets (power set) of a given finite set and comprehend the partial-order relationship between these subsets with respect to inclusion (for example, given the set {a, b, c} the corresponding power set is {Ø, {a}, {b}, {c }, {a, b }, {b, c}, {a, c }, {a, b , c}}.)

They test the validity of statements formed by the use of the connectives and, or, not, and the quantifiers none, some and all , (for example, ‘some natural numbers can be expressed as the sum of two squares’). They apply these to the specification of sets defined in terms of one or two attributes, and to searches in data-bases.

Students apply the commutative, associative, and distributive properties in mental and written computation (for example, 24 × 60 can be calculated as 20 × 60 + 4 × 60 or as 12 × 12 × 10). They use exponent laws for multiplication and division of power terms (for example 2³ × 2⁵ = 2 ⁸, 2⁰ = 1, 2³ ÷ 2⁵ = 2^-2 , (5²)³ = 5⁶ and (3 × 4)² = 3 ² × 4²).

Students generalise from perfect square and difference of two square number patterns
(for example, 25² = (20 + 5)² = 400 + 2 × (100) + 25 = 625. And 35 × 25 = (30 + 5) (30 - 5) = 900 - 25 = 875)

Students recognise and apply simple geometric transformations of the plane such as translation, reflection, rotation and dilation and combinations of the above, including their inverses.

They identify the identity element and inverse of rational numbers for the operations of addition and multiplication
(for example,  ½+ - ½ = 0 and ² / ₃ × ³ /₂ = 1).

Students use inverses to rearrange simple mensuration formulas, and to find equivalent algebraic expressions
(for example, if P = 2L + 2W, then W = ^P / ₂ - L . If A = p r² then r = v^A /p).

They solve simple equations (for example, 5 x + 7 = 23, 1.4x - 1.6 = 8.3, and 4x ² - 3 = 13) using tables, graphs and inverse operations. They recognise and use inequality symbols. They solve simple inequalities such as y = 2x+ 4 and decide whether inequalities such as x ² > 2y are satisfied or not for specific values of x and y.

Students identify a function as a one-to-one correspondence or a many-to-one correspondence between two sets. They represent a function by a table of values, a graph, and by a rule. They describe and specify the independent variable of a function and its domain, and the dependent variable and its range. They construct tables of values and graphs for linear functions. They use linear and other functions such as f( x) = 2x - 4, xy = 24, y = 2^x and y = x² - 3 to model various situations.

• Structure of algebraic expressions
  
  
• The meaning of letters in algebra
  
  
• Sets
  
  
• Manipulating symbols

Student express relations between sets using membership, ?, complement, ', intersection, n, union, ?, and subset, ?, for up to three sets. They represent a universal set as the disjoint union of intersections of up to three sets and their complements, and illustrate this using a tree diagram, venn diagram or karnaugh map.

Students form and test mathematical conjectures; for example, ‘What relationship holds between the lengths of the three sides of a triangle?’

They use irrational numbers such as, p, f and common surds in calculations in both exact and approximate form.

Students apply the algebraic properties (closure, associative, commutative, identity, inverse and distributive) to computation with number, to rearrange formulas, rearrange and simplify algebraic expressions involving real variables. They verify the equivalence or otherwise of algebraic expressions (linear, square, cube, exponent, and reciprocal,
(for example, 4 x - 8 = 2(2x - 4) = 4(x - 2); (2a - 3)² = 4a² - 12a + 9; (3w)³ = 27w³ ; (x³y) / _xy² = x²y ^-1; ⁴ /_{x y} = ²/_x × ²/_y).

Students identify and represent linear, quadratic and exponential functions by table, rule and graph (all four quadrants of the Cartesian coordinate system) with consideration of independent and dependent variables, domain and range. They distinguish between these types of functions by testing for constant first difference, constant second difference or constant ratio between consecutive terms (for example, to distinguish between the functions described by the sets of ordered pairs
{(1, 2), (2, 4), (3, 6), (4, 8) …}; {(1, 2), (2, 4), (3, 8), (4, 14) …}; and {(1, 2), (2, 4), (3, 8), (4, 16) …}). They use and interpret the functions in modelling a range of contexts.

They recognise and explain the roles of the relevant constants in the relationships f(x) = a x + c, with reference to gradient and y axis intercept, f(x) = a (x + b)² + c and f(x ) = ca^x.

They solve equations of the form f(x) = k, where k is a real constant (for example, x(x + 5) = 100) and simultaneous linear equations in two variables (for example, {2x - 3y = -4 and 5x + 6y = 27} using algebraic, numerical (systematic guess, check and refine or bisection) and graphical methods.

• Conceptual growth for solving equations
  
  
• Exponential functions