Planning tool
Year levels
Strands
Expected level of development
Australian Curriculum Mathematics V9: AC9M10A01
Numeracy Progression: Number patterns and algebraic thinking: P9
At this level, students extend the distributive law to expanding the product of two binomials (ax + b)(cx + d) and the factorisation of nonmonic quadratic expressions with integer coefficients. Students practise algebraic manipulation involving quadratic and exponential expressions to develop fluency and confidence with their use, and to identify errors in algebra.
They continue to use concrete and visual tools (for example algebra tiles and area models, diagrams, digital interactives) to scaffold conceptual understanding. They practise application of the distributive laws to expand binomial products algebraically and apply methods for factorising quadratic expressions based on relations between coefficients. They develop their recognition and use of expanded, factorised (including perfect squares, and difference of perfect squares) and completed square forms of quadratic expression.
Students practise using exponents laws to simplify expressions involving products, quotients and powers of numbers and variables, to develop their fluency with a range of combinations of these forms of expression, including cases with negative integer exponents. They learn to explain their working using counting exponents from first principles and application of the exponent laws.
Teaching and learning summary:
 Revise the distributive law using examples of simple twodigit number multiplications and show how this can be extended to multiplying binomials.
 Revise expanding binomial products of the form
(x + a)(x + b).  Revise factorising monic quadratic expressions with integer roots, and the relation between expanding and factorising as reverse processes.
 Work though examples of approaches to expanding products of binomials such as (2x + 4)(x + 3) then proceeding to products of binomials such as
(3x – 1)(4x + 5).  Work through examples of different approaches to factorising nonmonic quadratic expressions and finding the completed square form for quadratic expressions.
 Revise the exponent laws and extend them to expressions involving products, quotients and powers of numbers and variables and combinations of these.
 Work through examples of simplifying exponential expressions, including those involving negative integer exponents.
 Solve simple equations involving exponential expressions with one variable where the solution is an integer.
Students:
 expand, factorise and simplify quadratic expressions with integer coefficients
 convert between expanded and completed square forms of quadratic expressions
 relate quadratic expressions to diagrams and corresponding algebraic methods
 simplify exponential expressions involving combinations of products, quotients and powers of numbers and variables with
integer exponents  solve simple equations involving exponential expressions in one variable that have an integer solution.
Some students may:
 not correctly apply the distributive property when expanding the product of two binomial expressions, such as simply missing out middle terms, incorrectly thinking that (x + 3)^{2} = x^{2} + 9 or that (2x – 4)^{2 }= 4x^{2} – 16.
 multiply rather than add exponents when simplifying exponential expressions involving multiplication.
 divide, rather than subtract exponents when simplifying exponential expressions involving division.
 confuse the power of an exponential expression with the product of two exponential expressions, such as thinking (3^{4})^{2} = 3^{4} × 3^{2} rather than 3^{8}.
 not realise that a pronumeral without the exponent showing always has an exponent of 1 (for example, 2 = 2^{1}, x = x^{1}) or that base integers with an exponent of 0 equals 1 (3^{0} = 1, x^{0} = 1).
 not realise that a negative exponent corresponds to the reciprocal of the corresponding positive exponent.
The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate and extend learning done at school or for home schooling.
Learning intentions
 We are learning to consolidate expanding the product of two binomial expressions, (ax + b)(cx + d).
 We are learning to consolidate factorising nonmonic quadratic expressions.
Why are we learning about this?
Quadratic functions and their rules can be used to model situations and solve problems involving curves, areas, patterns and movements.
Formulating and manipulating quadratic expressions where the coefficient of x^{2} is different from 1 covers the full range of quadratic functions.
What to do
Practise
Use an interactive problem solver or graphics calculator to practise expanding quadratic expressions such as (2x – 3)(4x + 5):
 Complete the problem by hand on paper.
 Check your solution by typing the expression (2x – 3)(4x + 5) into the digital tool.
 If your tool allows, view the solution steps to see if your working is correct.
Now test yourself!
Expand each of the following and check your solution.
 (4x + 3)(5x + 5)
 (8x + 2)(7x – 2)
 (5x – 6)(7x + 4)
 (9x – 4)(3x – 5)
Try this
A group of students was asked to expand (3x + 4)(2x – 5) and between them they obtained four different answers, listed below.
Identify the three incorrect answers and explain the common error made in each case:
 6x^{2} – 20
 6x^{2} – 7x – 20
 6x^{2} + 8x – 20
 6x^{2} + 23x – 20
Practise
 Use a digital problem solver or graphics calculator to practise factorising quadratic expressions such as 3x^{2} – 10x + 8
 Complete the problem by hand on paper.
 Check your solution by typing the expression 3x^{2} – 10x + 8 into the digital tool.
 If your tool allows, view the solution steps to see if your working is correct.
You try!
Factorise each of the following and check your solution:
 2x^{2} + 11x + 5
 6x^{2} + 10x – 4
 8x^{2} – 20x + 12
 10x^{2} – 11x – 6
Success criteria – quadratic expressions
 I can expand the product of two binomial expressions.
 I can factorise quadratic expressions where the coefficient of x^{2} is different from 1.
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Teaching strategies
A collection of evidencebased teaching strategies applicable to this topic. Note we have not included an exhaustive list and acknowledge that some strategies such as differentiation apply to all topics. The selected teaching strategies are suggested as particularly relevant, however you may decide to include other strategies as well.

Explicit teaching
Explicit teaching is about making the learning intentions and success criteria clear, with the teacher using examples and working though problems, setting relevant learning tasks and checking student understanding and providing feedback.
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Worked examples
A worked example is not just a preworked question that is given to the students. There are several types of worked examples and ways of using them.
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Concrete, Representational, Abstract (CRA)
The CRA model is a threephased approach where students move from concrete or virtual manipulatives, to making visual representations and on to using symbolic notation.
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Teaching resources
A range of resources to support you to build your student's understanding of these concepts, their skills and procedures. The resources incorporate a variety of teaching strategies.

Factorisation
This teacher resource is a thorough resource that provides guidance on all aspects of factorisation at this level. It points to assumed knowledge required for this topic, factorisation approaches, worked examples, exercises and answers.
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Expand and factorise expressions
This teacher resource includes a range of activities related to expanding and factorising quadratic expressions.
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When and why do we write quadratic equations?
This page presents several different quadratics for students to solve. Students are stepped through worked solutions of exercises with increasing difficulty. Further questions are provided for students to practise completing quadratic equations.
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Expanding double brackets
This is an interactive rectangle that can be manipulated by units to allow students repeted practice as well as incremental practice expanding algebraic expressions. This allows students to visualise, make connections and identify emerging patterns.
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Factorising: numeracy workshop
This downloadable resource is a collection of PowerPoint slides that outline factorising from the beginning steps to polynomials.
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