Expected level of development
Australian Curriculum Mathematics V9: AC9M9A05
Numeracy Progression: Number patterns and algebraic thinking: P9
At this level, students take a mathematical-modelling approach with algebraic rules, tables, graphs and digital tools to construct linear and quadratic models for a range of real-life situations, make predictions and solve related problems. They spend time identifying variables to formulate the models and investigate the potential application of linear and quadratic models to some new situations. Investigations often require a combination of creativity and lateral thinking, scaffolded by an analytical and systematic approach.
Students are supported to use a variety of approaches to investigate a problem and strengthen their understanding of the connections between the rule, table and graph. Some useful real-world contexts for investigating relationships around change include speed, financial maths, paths of projectiles, parabolic mirrors, cooking time and temperature.
Students interpret results and communicate their method and findings to their peers, including any assumptions made, and limitations of their models, in relation to the context. They consider and assess feedback received, and refine their model accordingly, as required.
Teaching and learning summary:
- Have students investigate a range of real-world problems to which they can apply a mathematical-modelling approach.
- Encourage students to formulate their own problems, identify variables and create a model to develop solutions and make predictions.
- Have students communicate their method and results clearly and efficiently.
- Encourage refinement of the model, taking into consideration questions that may have come up during feedback. For example, have the students considered the most suitable model? What assumptions did they make? What limitations does the model have?
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 intention
- I am learning to identify the variables, constant rate of change and starting value used to construct a linear model.
- I am learning to apply a linear model to make predictions and solve real-world problems.
Why are we learning about this?
Modelling real-world contexts using mathematical functions allows us to better understand a situation, make informed decisions, explore alternative scenarios and consider future possibilities.
What to do
Representing trade quotes as linear functions
A plumber charges $120 per hour and an $80 call-out fee for a plumbing job.
- Complete the table of values below.
- Determine the rule that relates the cost of a job to the time taken to complete it.
- Graph this rule, either by hand or using graphing software such as Desmos or GeoGebra. Consider what scales to use for the horizontal and vertical axes.
| Time (hours) | 0 | 1 | 2 | 3 | 4 | 5 |
| Cost ($) |
Multiple functions
The graph below shows three different plumber quotes (Plumber A, Plumber B and Plumber C), represented as linear functions.
- Use the diagram to determine which plumber will cost the least for a 1-hour job.
- Which plumber is the most economical to use for a longer job that takes 3 hours or more?
- Consider Plumber A. How do their fees compare to the fees of Plumber B and Plumber C for different job times? Give reasons for your answer.
- A 3-hour job with Plumber B cost $360. Write a rule that represents Plumber B’s service fees.
- Use your rule (from question 4, above) to calculate Plumber B’s cost for a 1.5-hour job. Check your answer against the cost of Plumber B from the graph.
Success criteria
- I can identify the variables, constant rate of change and starting value to construct a linear model.
- I can apply a linear model to make predictions and solve real-world problems.
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Teaching strategies
A collection of evidence-based 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.
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Mathematics investigation
By giving students meaningful problems to solve they are engaged and can apply their learning, thereby deepening their understanding.
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Questioning
A culture of questioning should be encouraged and students should be comfortable to ask for clarification when they do not understand.
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Concrete, Representational, Abstract (CRA)
The CRA model is a three-phased approach where students move from concrete or virtual manipulatives, to making visual representations and on to using symbolic notation.
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Differentiation
Differentiation involves teachers creating lessons that are accessible and challenging for all students.
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Worked examples
A worked example is not just a pre-worked question that is given to the students. There are several types of worked examples and ways of using them.
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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.
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Writing equations to model relationships (part 1)
Students investigate quantity and variance when variables are given and when variables are not given. There is a level of repetition in the investigation to allow for practice and confident interpretation.
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Mathematical modelling
Use this guide to learn more about mathematical modelling and how it can be used in the classroom. Go to page 16 for a specific example in relation to linear functions to find the answer to a hypothetical question regarding Mozart and his opus.
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Real-world algebra: linear functions
Students use dynamic geometry software to investigate the variables of the linear function rule as line graphs are matched to features of object in an image to assess their suitability, for example, the gradient of a slide.
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Mathematical modelling: pricing for profit
Students use prior knowledge to model financial problems around profit and pricing. This activity presents problems from Year 7 to Year 9, providing opportunity for differentiation.
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Assessment
By the end of Year 9, students can use mathematical modelling to solve problems involving change in financial and other applied contexts, choosing to use linear and quadratic functions.
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Planning a pizza party
This problem provides prompts and questions for students to use linear functions to model problems when planning catering for a party.
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Enclosing squares
Set this task as a challenge. Students may work collaboratively to use their knowledge of algebra, Pythagoras’ theorem and graphing to explore and practise gradients and straight lines.
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Handshake problem
Students work in pairs or groups to nut out this problem. Students are required to apply a numerical model to a party scenario. They are asked to question their method in solving the problem – was it the most efficient approach?
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