Expressions, formulas and substitution

Curriculum information

Achievement standard

Students use algebraic expressions to represent situations, describe the relationships between variables from authentic data and substitute values into formulas to determine unknown values.

Content descriptions

Students recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown. AC9M7A01

Students formulate algebraic expressions using constants, variables, operations and brackets. AC9M7A02

General capabilities

Numeracy:

Number patterns and algebraic thinking (Level 6)

Critical and creative Thinking:

Interpret concepts and problems (Level 5)

Assessment

Use the following formative assessment to gauge where students are at.

Exit ticket questions:

If x = 2, find the value of 2x–9
Label the parts of this expression: 3x–2
In your own words, what is the difference between an expression and an equation?

Areas of challenge

Some students may:

think that the '=' sign indicates an example, rather than expressing an equivalence relationship
think that a variable represents the count for the number of objects (for example, 3a means 3 apples)
have difficulties understanding the lack of multiplication signs in algebraic expressions, such as 3 x a = 3a
have difficulties with different decimal and fraction representations in algebraic expressions, such as 0.5a = 1 2 a = a 2
think that a particular variable always holds the same value
think that the same variable in an expression can hold different values, such as thinking that t could be 2 and 4 in the expression t + t = 6.

Prerequisite student knowledge and language

Prior to this lesson, it is assumed that students have knowledge of:

the concept of variables
constructing algebraic expressions from word problems
conventions associated with the order of operations for integers
knowledge of algebraic conventions (covered in prior lesson)
how ‘=’ indicates an equivalence statement
what is meant by an expression (from prior lesson)
how to measure length accurately using rulers (to the mm)
how to use formulas to find the area of a triangle and rectangle
decimal notation and rounding decimals
using calculators to carry simple calculations
how to measure mass in grams.

Suggested strategies

Concrete, Representational, Abstract

Worked examples

Mathematical investigation

What you need:

Lesson plan (Word)

Teacher's slides (PowerPoint)

Warm up with algebraic expressions worksheet (Word)

Workstation instructions (Word)

Stacking cups and two different coloured counters, kitchen scale, rulers and calculators

Learning goals

Learning intention

We are learning to match word problems to their algebraic expression.
We are learning to understand the difference between an expression and a formula and learn algebra terminology such as constant, coefficient, variable and operator.
We will substitute values into equations in a range of scenarios.

Success criteria

By the end of the lesson, students can:

explain the difference between a variable and an expression
identity a variable, coefficient, constant and operator
substitute values into a range of equations in a range of scenarios.

Why are we learning about this?

Equations are useful in many everyday life scenarios – including representing our wages – or in formulas for finding the area of a shape or the density or volume of an object. Substituting values into equations allows us to easily explore a range of phenomena in the real world and solve a wide variety of problems.

Learning hook 15 mins

To start

Print the Warm up with algebraic expressions worksheet (one per student) and access the downloadable teacher’s slides to present this lesson. Have students complete the worksheet individually. Once complete, students can swap worksheets with a partner and compare answers. Take note of discussions between pairs and use this time to note misconceptions or concepts requiring more attention.

Introduce new concepts

Use slides 1 and 2 and refer to teaching notes provided with the slides to introduce students to the learning intentions of the lesson. Remind them to think back to what was learnt about constructing algebraic expressions. Explicitly explain you will be continuing to construct algebraic expressions and will be working with algebraic equations and substituting values to find and unknown (slides 3–5). As the term ‘equation’ is being introduced, explain to students the difference between equations and expressions, and discuss key terminology associated with expressions.
Show students the following example:

‘Mark gets $3 less pocket money a week than his older brother Sam.’

With the class, construct an equation that represents the scenario and use cups and counters to model this equation (slides 6 and 7). Explain to students that if we have an algebraic equation and we are told the value of an unknown, we can substitute this value into the equation. Use the example from above, substituting in that Sam gets $5 each week. Model the process of substituting in $5 for S by filling the cup with 5 counters. Show students how the value of the counters in the cup (+5) and the –3 equate to an answer of $2. Explain to students that in the question we have substituted S for 5 and that in algebra substitution this means replacing a variable with a particular value.

Go to slides 8 and 9 to conduct another whole-class example of substitution using cups and counters using the following ‘Taj has several books. Lulu has twice as many. Construct an expression to show how many books Lulu has. If Taj has 6 books, how many does Lulu have?’
Now work through another whole-class example without concrete materials (slides 10 and 11): ‘Roshini has a little brother, Rami. If you double Rami’s age and add 3 more years, you get Roshini’s age. If Rami is 3, how old is Roshini?’

Explore 35 mins

Enable students to explore the concept

Download and print the Workstation instructions to be displayed on tables for students to rotate through. Pre-cut the triangles and shapes ready to go. Arrange students in groups of three or four at each workstation. Allow students to engage in each task for 15 minutes, then rotate.

Workstation 1: Students measure the dimensions of a variety of cut-out rectangles and triangles using rulers. Using the formulas for the area of a triangle and area of a rectangle, students substitute measurements into the formula to calculate the area of each shape.

Workstation 2: Students measure the mass of various objects using kitchen scales. Using the formula for density, and each objects’ approximated volume, students calculate the density of each object and order them from least to most dense.

Workstation 3: Students consider a wage scenario and, with scaffolding, construct the algebraic equation to represent the total wages earnt each week. From this equation, students answer various salary questions to eventually determine Milly’s annual salary.

Workstation 4: Students consider the given graph and scenario: growth of a rabbit population on Gadigal land in NSW. In this scenario, students use the given equation that models the graph to extrapolate expected values for the rabbit population and will answer a set of given questions for the scenario.

Differentiation (all abilities): Each workstation has three differentiated levels of challenge: ‘mild’, ‘medium’ and ‘spicy’. Allow groups to choose the difficulty level at each workstation. It is encouraged that students work in mixed-ability groups to support effective collaborative learning.

For further reading go to: The spread of rabbits throughout Australia.