The prompt is inspired by Mike Ollerton's booklet Learning and Teaching Mathematics Without a Textbook. His starting point is x + y = 7, which is then followed by a list of questions to investigate. In contrast, inquiry maths takes a step back in the learning process by encouraging students to come up with the questions. Moreover, the prompt y - x = 4 challenges students, in finding pairs of numbers that satisfy the equation, to face the concept of subtracting negative numbers. In my experience, students in year 7 quickly link x and y to the axes of a graph, proceed to find coordinate pairs, and plot them on a graph.
This inquiry builds students' confidence to suggest changes to the prompt. They readily come up with alternatives to the operation and y-intercept. If you want the class to focus more on the gradient, you could change the prompt to y - 2x = 2 or a similar equation.
This inquiry has led to the introduction of rearranging equations when, for example, a student notices that the lines produced by y - x = 4 and y = x + 4 are the same. At GCSE level, students have looked at the angle of elevation using trigonometry and developed questions about similar triangles (see mathematical notes).
A common misconception that arises during this inquiry is that "y cannot be less than 4." The student reasons that if the value of y is below 4, then subtracting another amount can only make the answer smaller and, hence, y - x can never equal 4 when y < 4. The idea of subtracting negative numbers can be introduced by using a table of x and y values that satisfy the equation (as illustrated on the left).
Another issue that can arise when students use the prompt relates to plotting coordinates from the equation. The prompt is presented with y and then x, yet students write a list of coordinate pairs with the value of x first and plot the x-coordinate first. This is a deliberate obstacle to slow students down and make them think about the relationship between x and y. However, when students are not used to inquiry, the teacher might want to facilitate the change in representation (from equation to graph) by using the prompt x - y = 4.
Note sheet (four quadrants)
Promethean flipchart download
Smartboard notebook download