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Coordinates inquiry

When students in lower secondary school see the prompt, they often recognise four coordinates. The prompt is designed to tackle misconceptions about coordinates, particularly those with negative values and those on the x and y axes. During the orientation phase of the inquiry, the teacher should ensure that students can plot coordinates in all four quadrants. 
After the class has attempted to plot the four coordinates on an x-y axis, the teacher could call on individuals to plot them on the board. If students have not already identified the points as vertices of a quadrilateral, the teacher draws lines between the coordinates. A discussion (which, when necessary, the teacher initiates and orchestrates) ensues about the properties of the shape:
  • Is the shape a square?
  • Are the angles right angles?
  • Are the sides equal in length?
  • Are the pairs of sides parallel?
Students (individually or in pairs) share their reasoning, often describing how a square has been 'tilted'. The teacher can introduce the idea of right-angled triangles on the sides of the square (see illustration below) to facilitate students' thinking.
Once the class has established that the shape is a square, the teacher might structure the inquiry by suggesting students follow one of the lines of inquiry below or might guide students by inviting them to choose from some or all of the lines. Alternatively, the teacher might decide to run a more open inquiry by asking for 
students' suggestions and, depending on the mathematical validity of the suggestions, allow individuals to pursue their own ideas. If the teacher opts for a guided or open inquiry, the regulatory cards help to ensure that each student is aware of the direction in which their inquiry is going.
Lines of inquiry
(1) Plot other shapes
Students draw triangles, other 
quadrilaterals and polygons with five sides and more. The teacher could set the constraint that the origin has to be inside the shape, thereby ensuring students practise finding coordinates in the four quadrants.
(2) Extend the square
Students extend the pattern by drawing more squares. They use a vector instruction to move each point in a different direction: right 4, up 2 (translates each point to the right); right 2, down 4 (down); left 4, down 2 (left); left 2, up 4 (up).
(3) Gradients of parallel and perpendicular lines
Students calculate the gradients of the four lines using right-angled triangles. They establish that the diagram shows two pairs of parallel lines because the lines in each pair have the same gradient. They notice (with the support of the teacher if required) that the gradient of a line perpendicular to another line is the negative reciprocal of the gradient of that other line.
(4) Find the relationship between x and y for each line

The coordinates of the line that form the top line of the square are (-4,2) and (0,4). For the extended line, the coordinates continue (4,6), (8,8), (12,10), (16,12) and so on. The x-coordinate increases by four each time and the y-coordinate by two. To find the y- coordinate for each value of x, halve the x-coordinate and add four. This could be presented in the general form as (n,1/2n + 4) or as a number machine or an equation (see right).
(5) Generalisation

Use the coordinates in the prompt to form a generalisation. If you substitute a different value into the general form, do you still create a square? Create a different general form for four coordinates. Can you find four coordinates that create different shapes when you substitute different values into the terms?


Structured inquiry   
Michael Fenton
has devised a structured inquiry about coordinates and the gradient of straight lines. “Point, Point, Slope” starts with this prompt (right) from which students are required to form two pairs of coordinates. Their aim is to create lines with the greatest and least possible gradients and one with a gradient as close to zero as possible. Michael has excluded zero and one from the possible values of x and y in order to increase the level
 of challenge. You can find extensions and lesson notes on Michael’s website
The simpler prompt (below left) might initiate a guided or even open inquiry in which students and the teacher co-construct inquiry pathways. Students
might set their own constraints, including the use of
negative numbers (with the consequent expansion of the inquiry into four quadrants). They might also change the prompt to include three coordinates that all lie on the same straight line or even four coordinates that lie on two parallel or two perpendicular lines. (To read about the differences between structured, guided, and open inquiries, click here).

Michael is a junior high and high school teacher in Fresno, California. He regularly speaks and runs workshops at conferences on mathematics education in the US. You can follow Michael on twitter @mjfenton.
Classroom inquiry
Caitriona Martin devised the coordinates prompt below to use with her year 9 class. Here she describes how the inquiry developed:
This inquiry arose from the need to have a second inquiry lesson on straight line graphs, having already used the inquiry prompt
y - x = 4 during the previous ‘FIG Friday’ lesson. As we’re now following a mastery style scheme of work, we’re still on the same topic two weeks later, which feels like a really good thing as my year 9 class seem to need the time to be able to make the meaningful connections in this rich topic. Here are their questions and comments:
At a basic level, two groups just noticed which coordinates were positive and negative – these groups, with a little more thinking time, could possibly have gone on to ask the question, “Would the x coordinate always be positive and the y coordinate negative?”
The context within which this prompt was used meant that pupils were already familiar with the terminology associated with straight line graphs and, therefore, pupils were eager to apply this knowledge. The most common questions referred to finding the y-intercept and gradient of the line that joins the coordinates – definitely a task which stretched their ability.
One group wanted to add more coordinates to those two to make a shape. This has the potential to be very easy or very challenging indeed. I was surprised when 4 out of the 8 groups in my class chose this question for their inquiry, so I encouraged them to work out the gradients of the line segments which they used for their shape in order to ensure they were still doing maths that would challenge them. (This pathway offers the potential to lead to parallel and perpendicular lines if rectangles were the shape of choice.) 
Another group wanted to draw a circle and did attempt this, although in hindsight I would have guided them more to join the coordinates first and use them as a diameter, so that they might get more out of it mathematically. The groups that got on well with the task plotted the coordinates on a graph and wrote down the coordinates in a table to look for the sequence in the y-coordinates.

Caitriona was second-in-charge of the maths department at St. Andrew's School, Leatherhead (UK) at the time of devising the prompt. She introduced 'FIG Fridays' to promote functional and inquiry maths, as well as groupwork. You can follow her on twitter @MrsMartinMaths.