Conceptual Growth for Solving Equations - Progression Points

Dimension

Level

Progression Point

Structure

2.25

• Use of ‘=’ to indicate equivalence or the result of a computation

2.5

• Construction of number sentences
• Calculations using notation such as ‘3 + 5 − 2 =’

3.0 Standard

… Students understand the meaning of the ‘=’ in mathematical statements and technology displays (for example, to indicate either the result of a computation or equivalence).

They construct number sentences with missing numbers and solve them.

3.25

• Use of trial and error to find a missing number in a number sentence; for example, 4 × ? + 6 = 22

3.5

• Consistent and correct use of conventions for order of operations

4.0 Standard

… Students solve equations by trial and error.

4.5

• Use of inequality, equality, approximately equal and not equal, including in symbolic expressions
• Solution of simple linear equations using tables, graphs and inverse operations (backtracking)

4.75

• Solution of equations such as x² = 17 as required in measurement situations; for example, using pythagoras theorem

5.0 Standard

… 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 = πr2 then r = √A/π for r > 0).

They solve simple equations (for example, 5x + 7 = 23, 1.4x − 1.6 = 8.3, and 4x2 − 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 x2 > 2y are satisfied or not for specific values of x and y.

5.25

• Solution of equations by graphical methods

5.5

• Expansion of products of algebraic factors, for example, (2x + 1)(x − 5) = 2x² − 9x − 5
• Use of inverse operations to re-arrange formulas to change the subject of a formula

5.75

• Factorisation of simple quadratic expressions and use of the null factor law for solution of equations
• Formulation of pairs of simultaneous equations and their graphical solution

6.0 Standard

… Students 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.