Solving Linear Inequalities

Semester 2 has started, which means the Computer Science elective I was teaching has ended. The good news is, I’ve taken over Year 10 Maths, which means my teaching load is more maths than it’s ever been before. I haven’t taught Year 10 for a couple of years now, and I’ve changed the way I teach a lot even in the last 12 months. Luckily, I had this same class last year so they’re pretty used to how I do things.

The first unit I’m teaching is Linear Relationships:

LR skill list

Most recently we’ve been working on LR2, linear inequalities. If these are taught as a totally procedural matter, it’s a fairly easy topic: solve them the same way as equations, just being careful with the direction of <, >, ≤ or ≥ if dealing with negatives. But as I always tell my students, I believe our aim is not to ‘get the right answer’, but to understand.

In particular, I want my students to understand that there is not just a single solution, but a whole set of solutions. I want them to understand that when we write a statement such as x ≥ -2, we are describing a rule by which some values are included and some are not. So I started the lesson by looking at a couple of examples:

A little is lost when seeing this as a static IWB page, as opposed to the notes that developed through class discussion. Importantly, all the possible solutions were provided by students. I was really pleased with their suggestions, in that they illustrated some important points about inequalities. For example, 4.999999999 is indeed less than five, as are all negative numbers. And I liked the suggestion of 6000893, pointing out that there isn’t an upper limit for x ≥ -2. And I impressed the student by hearing and remembering the number he called out :).

Next I gave an example of an inequality to solve. Rather than getting the class to solve it as an equation, I had them each make a list of five values that would be included in the solution, and five that are not. We then shared some of these as a class:

What this allowed us to do was find the solution to the equation by understanding what the equation described. It’s important as a maths teacher to ensure students know why they do things they do. When students solve equations, they shouldn’t following a list a pre-described steps to get ‘the answer’. They should be trying to answer the question, “What value makes this statement true?”

We also looked at 23 – 2x ≥ 15, and found that our solution doesn’t necessarily have the same direction as the inequality. My students quickly identified the negative in front of the x as the culprit. As one student pointed out, you need to make x smaller if you want 23 – 2x to get larger.

So, the class found two important pieces of information are needed to find the solution to an inequality:

  1. What is the boundary between the solutions that are included and the solutions that are not?
  2. Which direction do those solutions go in?

And how do we find this information? Using algebra, of course!

It’s at this point that I believe procedure becomes useful: after that understanding has been built. I’m increasingly finding that a good way to get students to follow a process I give them is to use a less efficient method they understand to do the same thing first. Students could solve inequalities by finding test cases each time, but they find solving them algebraically is a much quicker process. They look for shortcuts to make their work easier. Just like mathematicians do.

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