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Coordinates inquiry

The coordinates prompt was designed by Caitriona Martin (see box). It has the potential to develop an understanding of:
  • Plotting coordinates in four quadrants;
  • The gradient of a line (and parallel and perpendicular lines);
  • The y-intercept;
  • Coordinate shapes when the coordinates are viewed as vertices;
  • Other coordinates on the line that joins the two coordinates.
The gradient of the line that joins the two coordinates is -1/2, which could be challenging for some classes. A more appropriate prompt to introduce graphs of linear equations might use two coordinates that have a clearer connection between the x- and y-coordinates. For example, in (2,5) and (4,7) the y-coordinate is three greater than the x-coordinate. Thus the equation of the line is y = x + 3 and other coordinates on the line are of the form (n,+ 3). Similarly, the coordinates (-1,3) and (3,11) are on the line y = 2x + 5 and have the general form (n,2n + 5). Another option for the prompt is to give three coordinates from which students would be able to infer a generalisation more securely. 
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.

Promethean flipchart    download
Smartboard notebook   download

The genesis of the prompt
Caitriona Martin explains the development of the coordinates inquiry:
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.) 
One 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.