Graphing inequalities inquiry

The prompt

Mathematical inquiry processes: Verify; explore; set constraints; generate more examples. Conceptual field of inquiry: Plot straight lines; represent inequalities graphically.

The inquiry starts with students' questions and observations. Here are some examples that have arisen in classroom inquiry:

The teacher might introduce the statement in the prompt with the diagram below. The inquiry would then start with students trying to make sense of the two stimuli together. However, the teacher might choose to co-construct the diagram by drawing on students' prior knowledge. In either case, the teacher should ensure the class is aware of the difference between dashed and solid lines.

Whether the teacher presents or co-constructs the graph, students can use it to verify that the prompt is true. The only point (using integer coordinates) that satisfies the inequalities is (4,3).

The prompt piques students' curiosity and leads them to challenge themselves to find, for example, inequalities that define regions with different numbers of points. Students can set aims from their initial questions or, in a structured inquiry, the teacher might direct students towards a line or lines of inquiry from the suggestions below.

A line of inquiry

Is there always one point in the regions that satisfy the inequalities with consecutive whole numbers? What if we changed the inequalities to y < 5, x < 6 and x + y > 7? Or y < 6, x < 7 and x + y > 8

One student conjectured that if the numbers are consecutive, then there would always be one point in the region that satisfies the inequalities. Even though the conjecture turned out to be false, the numbers of points increase in a pattern. They turn out to be consecutive triangular numbers.

This could be expressed algebraically. For y < n + 3, x < n + 4, and x + y > n + 5, the number of points is 1/2n(n + 1).

If we inquire into the case 'before' the inequalities in the prompt (y < 3, x < 4 and x + y > 5), we get a region without integer coordinates. The same happens for the next case, but with a smaller region. For y < 1, x < 2 and x + y > 3, the lines intersect at (2,1) and there is no region. We can form a region for the next case, however, if we reverse the inequalities: y > 0, x > 1 and x + y < 2.

Other lines of inquiry

1. Points in the region that satisfy the inequalities

Is it possible to make up three linear inequalities that enclose a region with two, three, four or more points?

2. Change the order

What if we used the same consecutive numbers, but in a different order? How many points, for example, are there in the region that satisfies the inequalities y < 6, x < 5, x + y > 4? 

3. Other shapes

Could we create inequalities that define a region in the shape of a rectangle, parallelogram or trapezium?

4. Connection between the number of points

There is only one point in the region that satisfies the inequalities y < 4, x < 5 and x + y > 6, but there are 10 points in the region that satisfies the inequalities y ≤ 4, x ≤ 5 and x + y ≥ 6. What is the relationship between the numbers of points when the inequalities do not include 'equals to' and when they do?

5. Quadratic inequalities

How many points are there in the region that satisfies the inequalities y > x2  and x + y < 2? Is it possible to make up inequalities (one linear and one quadratic) that enclose a region with a specified number of points? Is it possible to do the same if one of the inequalities is cubic? Students could be encouraged to explore using Desmos.

6. Linear programming: optimisation

The teacher might introduce the idea of linear programming. Students could find the greatest or lowest sum of x and y in the feasible region and even model a situation in which they use an objective function to maximise profit in the production of x and y.

Resources