{"id":269,"date":"2015-07-22T05:07:09","date_gmt":"2015-07-22T05:07:09","guid":{"rendered":"http:\/\/newblog.primefactorisation.com\/2015\/07\/22\/solving-linear-inequalities\/"},"modified":"2024-11-02T15:09:15","modified_gmt":"2024-11-02T20:09:15","slug":"solving-linear-inequalities","status":"publish","type":"post","link":"https:\/\/www.primefactorisation.com\/blog\/2015\/07\/22\/solving-linear-inequalities\/","title":{"rendered":"Solving Linear Inequalities"},"content":{"rendered":"

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.<\/p>\n

The first unit I’m teaching is Linear Relationships:<\/p>\n

\"LR<\/p>\n

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.<\/p>\n

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:<\/p>\n

\"\"<\/p>\n

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 :).<\/p>\n

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:<\/p>\n

\"\"<\/p>\n

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?”<\/p>\n

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.<\/p>\n

So, the class found two important pieces of information are needed to find the solution to an inequality:<\/p>\n

    \n
  1. What is the boundary between the solutions that are included and the solutions that are not? <\/li>\n
  2. Which direction do those solutions go in?<\/li>\n<\/ol>\n

    And how do we find this information? Using algebra, of course!<\/p>\n

    \"\"<\/p>\n

    It’s at this point that I believe procedure becomes useful: after<\/em> 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.<\/p>\n","protected":false},"excerpt":{"rendered":"

    We’ve been working on linear inequalities. If these are taught procedurally, it’s a fairly easy topic. But our aim is not to ‘get the right answer’, but to understand.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-269","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/posts\/269","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/comments?post=269"}],"version-history":[{"count":1,"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/posts\/269\/revisions"}],"predecessor-version":[{"id":750,"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/posts\/269\/revisions\/750"}],"wp:attachment":[{"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/media?parent=269"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/categories?post=269"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.primefactorisation.com\/blog\/wp-json\/wp\/v2\/tags?post=269"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}