Expected level of development
Australian Curriculum Mathematics V9: AC9M7A04, AC9M7A05, AC9M7A06
Numeracy Progression: Measuring time: P7, Interpreting and representing data: P6, Number patterns and algebraic thinking: P7, P8
At this level, students are introduced to the four-quadrant Cartesian plane. The format of the coordinates should be explained and demonstrated; coordinates do not have to be an integer value. Give students time to draw axes and plot points so that they can familiarise themselves with the meaning of coordinates. Give helpful hints: x comes before y in the alphabet so plot x before y (x, y).
It is important that students are familiar with and confident in using the language of coordinate geometry. An understanding of the terms below is essential for students: x-axis and y-axis; x-coordinate and y-coordinate; coordinate pair; x-intercept and y-intercept; gradient; origin, quadrants; positive and negative direction, vertical and horizontal.
Students are shown how to use graphs to investigate and interpret real-life data such as travel graphs that investigate distances travelled or temperature graphs that compare weather at different times or the number of steps taken at different days of the week.
Being able to interpret graphs is an important life skill and students should be directed to focus on the meaning of the axes, the lines and the gradient (slope) of the lines.
To facilitate the transition from a measurement context to an algebraic one, the use of tables to record a series of points should be demonstrated and students should be encouraged to look for patterns in these tables. This is a good introduction to solving simple linear equations. To extend students, ask them to consider how applying different operations to one or both coordinates transforms its position on the plane.
Teaching and learning summary:
- Introduce the language of Cartesian coordinates.
- Give students multiple opportunities to plot and identify coordinates.
- Show students examples of real data plotted on a graph and discuss the meaning of each element.
- Start to make the links between a graph and linear algebra using tables and patterns.
- Introduce the use of digital tools, such as Excel, for recording data and plotting graphs.
- Guide students to consider how real-world data might be interpreted to be meaningful or misleading.
- identify and label the elements on a Cartesian plane
- identify a given coordinate pair on a graph
- plot a given coordinate pair on a graph
- correctly write a coordinate pair
- make a table of values from a set of coordinate pairs and identify patterns in the table
- interpret the meaning of a graph of data such as distance/time and temperature/time
- draw a graph that illustrates data, such as distance/time or temperature/time
- use a digital tool, such as Excel, to record data and plot a graph.
Some students may:
- mix up the x- and y-axes.
- write the coordinates with the y-coordinate first.
- move up/down before right/left.
- label the quadrants clockwise instead of anticlockwise or label the wrong quadrant first.
- not connect the left as negative and the right as positive.
- not recognise that the axes are not located in any quadrant, for example, the point (2, 0) is not located in Q1, it is on the x-axis.
- fail to connect how the coordinates represent real-world data and what that means.
Students may have difficulty making the connection between the graph as a picture and the relationships it represents. Using 'storytelling’ to explain what is happening in a graph is a good way for students to make sense of this type of data.
During this topic, begin classes with low-stake activities for students to practise only plotting points and recognising quadrants. For example, generate a set of points that will form a dot-to-dot image when plotted. Include questions such as 'In which quadrant is the image located?'
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.
- We are learning how to display numerical information on the Cartesian plane.
- We are learning how to analyse a relationship by observing features of a graph.
Why are we learning about this?
Almost anything in our world can be graphed, and these graphical pictures can tell us much more than numbers alone. Are the flood waters rising or falling? How quickly? Is there time to evacuate? When we visualise numerical information, we learn many things about it, such as understanding the past or predicting the future.
What to do
We are going to play a game of ‘guess my code’. To play this game, you will create a secret code by combining any two mathematical operations (+, −, ×, ÷), for example, multiply by 3 and then subtract 4.
Then you will apply the secret code to some of the numbers in the table below, but leave some cells blank.
Remember, order is very important in maths. You might like to see what happens if you apply your code in the opposite order.
This example that uses the sample code mentioned above (multiply by 3 and then subtract 4).
0 × 3 = 0
0 + 4 = 4
3 × 3 = 9
9 + 4 = 13
Part A: your turn!
- Give your partially filled table to another member of your household and see if they can guess the missing numbers and determine your code. If they crack the code, ask them how they did it. What strategies did they use to determine the relationship between the numbers before and after the code? Are there any patterns?
- Create a new code and fill in a new table. However, this time do not give it to your ‘guesser’; that is, instead, plot your ‘before’ and ‘after’ numbers as points on a Cartesian plane. To explain, using the example above, you would plot the points (–2, –2) in the third quadrant, (0, 4) on the y-axis, and (3, 13) in the first quadrant. Remember that the x-coordinate represents the number before the code was applied and the y-coordinate represents the number after the code is applied.
- Show your plotted points to your guesser and see if they can determine the points that are missing.
There should be seven in total, so your x-axis will need to run from –3 to +3.
- While your guesser is working, notice whether your points are sloping up from left to right or sloping down. Why is this happening? See if you can create a new code with the opposite slope.
- Now discuss with your guesser whether they found it easier to guess using the table or using the Cartesian plane. Why?
- Now try playing in reverse. Ask your guesser to create a code for you and see if you can break it with a table or plotted points.
Part B: make connections
Now we are going to try to write your code using mathematical symbols. (Remember those things called ‘variables.’)
- Let’s call the ‘before code’ number x and the ‘after code’ number y. Using the example above, convert your code to y = x × 3 + 4. Notice how we do not need to put brackets around the x × 3 because multiplication comes before addition in the mathematical order of operations.
- But we're not quite finished yet. Let’s tidy up the code. We do not need to include the multiplication sign (×) at all, because when we put a number next to a letter, an invisible × is created! So, the code is y= x3 + 4.
- Wait! That does not look right. One last step. We need to move the 3 in front of the x. This makes sure we are following the convention that mathematicians have used for hundreds of years. We should not spoil that tradition. So finally, y = 3x + 4.
- Can you convert all your codes into mathematical symbols (equations)? Try to spot a specific pattern.
- Study the tables you created and notice by how much the ‘after code’ values are increasing or decreasing with each step.
- Is there any link between this increase/decrease and a particular spot in your newly formed equations?
- I can create a set of ordered pairs with a linear relationship.
- I can plot these pairs on the Cartesian plane.
- I can determine whether my relationship is increasing or decreasing and compare how steep it is.
Please note: This site contains links to websites not controlled by the Australian Government or ESA. More information here.
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.
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.Go to resource
Using games and storybooks
Games and storybooks are great resources to use in the classroom and are engaging for students.Go to resource
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.Go to resource
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.
Practical numbers: Part 1
In this lesson, students explore standardised measuring systems.Go to resource
Graphs: formulas and variables
In this lesson, students use algebra and linear equations to model two real-world scenarios.Go to resource
Plotting points on the Cartesian plane
This is an exhaustive web page on this topic. Teacher guidance is detailed and includes practical advice to teachers, including interactive resource suggestions, activities and assessment ideas.Go to resource
This is an interactive coordinate plane to plot points, draw straight lines and measure slopes.Go to resource
This teacher and student interactive resource explores the slope of a straight line.Go to resource
Parallel lines in the coordinate plane: quick exploration
This interactive resource allows students to explore the properties of straight lines on the Cartesian plane, in particular, those that are parallel.Go to resource
By the end of Year 7, students use graphs to represent the relationships between variables. Students can describe functions that are derived from authentic data.