Planning tool

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 mathematical-modelling 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 dot-to-dot) 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:

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:

(If you like, categorise your data by gender for example, female, male, non-binary; 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 easy-to-read way:

• Have you labelled the axes?
• Does the graph have an informative title?
• Does the colour-scheme 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 bottom-left to top-right, that shows the overall trend of your data.

Check your line against a computer-generated 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.