Planning tool

Students will:

• identify an equation as linear
• understand what the variables in an equation represent
• form a table of values from an equation
• draw and label axes accurately
• plot coordinates from a table of values
• find the rule for a linear relationship – identify and calculate the gradient and y-intercept
• solve linear equations using a variety of techniques.

Students may:

• confuse linear graphs and statistical graphs. The notation used may be different for these. This needs to made clear to students and the differences discussed.
• miscomprehend 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.
• not understand that the linear function does not have to pass through the origin.
• have difficulty with a horizontally flat graph with a 0 gradient. It does not increase in an infinite interval. A vertically straight graph has an infinite gradient. It increases by an infinite amount in an instant.

It is beneficial to create a table of results when plotting a linear function. The coordinate pairs arise from the x and y values.

This resource shows teachers how to identify some of the misconceptions students may hold when solving linear equations and how to address them when teaching.

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 compare linear relationships by examining their gradients and intercepts.
• We are learning how to solve equations that answer a specific real-world question.
• We are applying our knowledge to make smart decisions and potentially save money on our electricity bill.

Why are we learning about this?

There are so many different options for everything these days. Even picking a soft drink at the shop can be overwhelming. It is important to use mathematical tools to compare our options and make a sensible decision. Linear relationships are very common, and although they might not help us decide which flavoured drink we feel like on a hot day, they can certainly help us to save money. 

What to do

Visit the government’s free energy comparison website Energy Made Easy . You might need help from someone at home to enter a few details if you want to be accurate for your own home. Otherwise, select a location and imagine a household of people and their energy usage. This website will then produce a list of different retailers sorted by price. To do this, the website is performing several invisible mathematical operations. Let’s dig into them a little deeper.

Select three plans that you want to compare. Click ‘view plan’ for each and note if there any fees or charges. In the example included here you can see this plan has a total of $58.98 (excluding the credit card payment fee). 

1. What is the cost per kilowatt/hour (kWh)?

Most retailers offer different rates for different times of the day; this website takes that into account. (You could try that as an extension task too but for now we want to simplify things.) Let’s select a price in the middle. I think 28c/kWh is appropriate here.

Combine your data to form a linear relationship. For this plan we have the following relationship:

Price = $58.98 + $0.28 × kWh

where price is the dependent variable and hours is the independent variable.

Repeat this for multiple plans. Try to find plans that have different offers. Compare plans with low fees and higher hourly rates against plans with higher fees but lower hourly rates. This will be more interesting in the next steps. It might be helpful to fill in a table like this:

Use GeoGebra to graph these relationships.

2. What do you notice about the graphs of the two relationships above?

3. Which has a steeper gradient? Can you predict what we would see if we scrolled to the right?

4. Which of these is a better deal?

5. Is there actually a correct answer to that last question?

6. Based on the comparisons you have made, which plan would be the most suitable to select?

Keep in mind we have over-simplified things a little. You probably need to answer this question in separate parts.

7. Would you suggest a different plan to a single person with a lower energy consumption versus a family with a much larger consumption?

8. At what point does one plan switch from being the more expensive option to being the cheaper option?

To find the exact point at which the plans ‘cross over’, we will need to solve a linear equation. Using the example above, I will put one relationship on the left and one on the right, with an equals sign between. Basically I am asking when these two plans are equal. Pairing that information with the picture above, we can provide the most accurate information.

0.28k + 58.98 = 0.31k + 51.13
 7.85 = 0.03k
        k ≅ 261.67

Make your final pitch. Present your graphs and equations to a member of your household and have a conversation about which energy retailer might be the best choice. Go back to the Energy Made Easy website and see whether your simplified model makes the same suggestion as the more complex algorithm of the website.

Success criteria

• I can use linear relationships and digital technologies to make informed decisions.
• I can solve equations, create plot graphs and interpret the information provided.