Mathematical inquiry processes: Extend patterns; generate examples; find relationships; generalise. Conceptual field of inquiry: The coordinate plane; gradients of parallel and perpendicular lines; coordinates and polygons.
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 ends of the line segment at the top of the square are (-4,2) and (0,4). If the line is extended rightwards, 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 below left).
Use the coordinates in the prompt to form a generalisation (see above right). 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?
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.
At the time of devising the prompt, Caitriona was second-in-charge of the mathematics department at St. Andrew's School, Leatherhead (UK). She introduced 'FIG Fridays' to promote functional and Inquiry Maths, as well as groupwork.