Planning tool
Year levels
Strands
Expected level of development
Australian Curriculum Mathematics V9: AC9M8A03
Numeracy Progression: Number patterns and algebraic thinking: P7
At this level, students apply a mathematicalmodelling process to linear relationships in realworld situations, such as financial contexts (for example, pay rates for parttime work) or measurements (for example, cooking, constant speed and distance/time graphs, leaking water tanks, sporting scenarios).
Students are creative and formulate problems using their knowledge and skills gained in this area. For instance, students may explore data and identify linear relationships between two variables. They formulate a problem, model it (perhaps using graphing tools), make interpretations and then communicate the appropriateness of their chosen model.
Ask questions to guide students in their mathematical exploration. How many coordinates do you need to accurately draw a linear function? Was the model chosen the most efficient way of exploring the linear relationship? Would there have been other, more efficient or appropriate ways to explore their chosen relationship?
Allow students to work collaboratively in their investigations. Encourage dialogue between classmates to come up with the best approach.
Teaching and learning summary:
 Revise the topic area to this point.
 Question and encourage discussion about the purpose and mathematicalmodelling approaches to linear relationships.
 Help students to identify linear relationships between two variables and how meaningful realworld data can be used to explore linear relationships.
 Use algebraic expressions and graphing knowledge to approach the investigation and modelling of realworld variables.
Students will:
 identify linear relationships between two meaningful variables that demonstrate a constant rate of change
 model this relationship and use algebra and graphing to explore this relationship
 model appropriately and efficiently in relation to a given relationship
 collaborate and then communicate their mathematicalmodelling efforts.
Students may:
 confuse linear graphs and statistical graphs. The notation used may be different between them. This needs to be made clear to students, and the differences discussed.
 not understand that the gradient can be calculated from any two points along the graph, not necessarily from the origin. Discuss that a line segment has the same properties as a complete line and that you don’t require many points to make a straight line.
 not realise that a linear function does not have to pass through the origin.
 not comprehend that it is beneficial to create a table of results when plotting a linear function. The coordinate pairs arise from the x and y values.
 not realise that y = mx + c can be adjusted to suit the model/context. For example, if we were measuring height versus age, the equation could be: h = ma + c.
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
 We are learning how to accurately collect and visualise bivariate data (data with two variables).
 We are learning how to build a model to represent our data.
Why are we learning about this?
It is extremely difficult to collect a complete set of data. That’s why our government only conducts a census every five years. The census is a great way to get accurate and comprehensive information, but it isn’t always vital – or possible – to collect such detailed data. Modelling allows us to answer questions and make predictions without a complete data set. Fitting a line or curve (like dottodot) through the points of data we do have, unveils information about the data that is missing. However, we must keep in mind that there will always be a margin of error with this method. There are ways we can mitigate that, though.
What to do
Build your own model and check its accuracy.
You will need:
 a tape measure
 paper and pens
 a ruler
 spreadsheet software (such as Microsoft Excel or Google Sheets).
Find as many people as you can and collect data on their age and height. Organise the data in a simple table like this:
Age  Height (cm)  Coordinates 
31  168  (31,168) 
Keep in mind that the more data you collect and the more variation in age it includes, the better your model will be. Your goal is to get it as close as possible to the true models you can find online.
Enter the first two columns of data into a spreadsheet. For Age vs Height, it should look something like this:
Age  Height 
31  168 
12  142 
1  81 
(If you like, categorise your data by gender for example, female, male, nonbinary; but you will need to add a further column.)
Using your mouse or trackpad, select all the data (including the headings) and insert a scatterplot. If you are using Microsoft Excel, click the ‘Insert’ tab and then tap the scatter button.
If you like, create multiple scatterplots that you can compare later. One should include all your data, and then create one that is specific to each gender.
Take some time to adjust the visual settings for your graph(s). Data is most useful when it is presented in an easytoread way:
 Have you labelled the axes?
 Does the graph have an informative title?
 Does the colourscheme add to the readability?
 Can you think of a way to distinguish the different graphs from one another, while keeping their core theme consistent?
Print your graph. (You could use a stylus or touchscreen instead.)
Study your graph(s) and try to identify any trends. Remember, we always work from left to right when finding relationships between variables.
 Is your scatter sloping up or down?
 Does the steepness of the slope increase or decrease at any point?
 List five things that you notice.
 If you made multiple graphs, compare and contrast your graphs. What do you notice?
Now it’s time to draw in a 'line of best fit'.
Try to draw a line through your scatterplot that highlights the trends you just identified. You might like to split it up into sections. If there is a section of your scatterplot that is obviously increasing, draw a line that slopes up through that section. If there is a separate section of your scatterplot that is flat, draw a flat line through it. If you draw in multiple lines, make sure they touch at the endpoints. You should also include a single line, from bottomleft to topright, that shows the overall trend of your data.
Check your line against a computergenerated one. Most spreadsheets allow you to add a trendline to your scatterplot. If you are using Excel, you will find this by clicking the + (chart elements) next to your graph. How close is the trendline to the line that you drew? How do they differ?
Look for holes in your scatterplot and then use the trendline to make predictions about groups of people you didn’t sample. Compare your scatterplot to more detailed ones online by googling images of ‘height age scatterplot’.
You will find some with trendlines included. Do they match yours? You will probably notice that they are not straight lines, like yours. As you continue your mathematical journey, you will be able to make your models more complex, by selecting a curve that is best suited to the scatterplot you see. Whenever you learn about a new type of graph, such as a parabola or sine wave, it is also important to consider what kinds of relationship modelling they might be useful for.
Success criteria
 I can collect data from a variety of people.
 I can organise data on paper and with technology.
 I can create a scatterplot to visualise data and insert a trendline.
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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.

Addition chain
This lesson is designed to consolidate skills in algebra, including collecting like terms, and expanding and factorising using the distributive law. Students use spreadsheets to investigate potential arithmetic relationships and then use algebra to identify and justify which relationships are generally true.
Go to resource 
Working with algebra
This resource includes three separate activities that consolidate the core algebra skills of operating with and manipulating algebraic expressions.
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Plotting linear relationships
From the AMSI Supporting Australian Mathematics Project, this resource gives teachers and students detailed explanations of the curriculum content, worked examples and assessment questions. This resource focuses on how to plot points on a Cartesian plane, and how to find the gradient and equation of a line.
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Linear and nonlinear relationships
In this resource, teachers are shown that by changing their questions they can guide students to successfully solve problems and understand new concepts. The topics covered include linear and nonlinear relationships. There are examples of classroom activities and teacher question stems. The activities are based on the BitL tool.
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Your number was ...
This problem prompts students to think mathematically and come up with a model to explain why the end result of this game always comes out as it does.
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Assessment
By the end of Year 8, students use mathematical modelling to solve problems using linear relations, interpreting and reviewing the model in context.

Working with algebra
This resource includes three separate activities that consolidate the core algebra skills of operating with and manipulating algebraic expressions.
Go to resource 
Addition chain
This lesson is designed to consolidate skills in algebra, including collecting like terms, and expanding and factorising using the distributive law. Students use spreadsheets to investigate potential arithmetic relationships and then use algebra to identify and justify which relationships are generally true.
Go to resource 
Your number was ...
This problem prompts students to think mathematically and come up with a model to explain why the end result of this game always comes out as it does.
Go to resource