In my own mind, there isn’t much sense in any of the inequality work that we do if we don’t use it to model real world scenarios. In this particular lesson, we look at classic problems involving “at least” and “at most.” These word problems allow students to set up inequalities and then solve them using the properties that we developed in earlier lessons.
In this lesson we look at classic interval notation. Although this is not officially part of any Common Core standard, nor clarified by New York State, the notation is helpful for students and I have found that many of them prefer the idea of simply specifying a starting and stopping value and then deciding whether they are included or not. It seems that the expectation, by New York, at least, is that students will know this notation heading into Common Core Algebra II.
In this lesson we work with a topic that can be, a huge source of confusion for students, that is graphing compound inequalities and the solutions to inequalities. We work on moving back and forth between two inequalities joined by the AND condition and a single inequality that indicates a portion of a number line.
In this lesson I’m looking to solidify students understanding of putting two inequalities together with the AND and OR conditions. This also sets the stage for understanding systems of equations and inequalities.
As strange as this may sound, this was one of my favorite lessons. I’ve taught solving linear inequalities for years, but I never really thought of the properties of inequalities. I knew them and of course knew the rule of switching the inequality sign when multiplying both sides (or dividing) by a negative. But, I had never framed it as a property issue, which it clearly is. So, I like that more technical basis of understanding the process. I don’t know if the kids will share my enthusiasm.
So, I wanted to make sure, first, that students understand that some inequalities are true and some are false. Then, I wanted to make sure that students really understood that a solution to an inequality, no matter how weird the inequality looks, is simply a value of x that makes the inequality true. I think many times our techniques for solving inequalities loses this basic truth.
O.k. This is definitely the lesson with the longest name. We used to call these literal equations back in the day, but now they are just linear equations with unspecified constants. Of course, what tend to show up on the New York State Regents exams could be linear, could be quadratic, you just never know. Still, this is a good lesson to stress the importance of reversing the actions done to the input variable in order to find its value.
In this lesson we tackle the very tricky topic of consecutive integer word problems. We all know they can be tested and the classic struggles students have with consecutive integers versus consecutive odds and evens are stressed. This lesson gives kids good practice on translating technical English to math.
In this lesson I try to cover the basics of modeling classic linear word problems in a single variable. This is a difficult skill for students and in all likelihood they will need another lesson. We do have a follow up with consecutive integers, but even more practice is likely needed.
Understandably, the Common Core standards ask for students to be able to justify each step in solving an equation by using either a property of real numbers, such as the commutative and associative properties, or a property of equality. I worry a bit about this devolving into another exercise in students memorizing the names of the properties, yet still not understanding why they are important.
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