I will begin with the essential question: How can we solve equations and inequalities with fraction coefficients?
In order to make sure students understand the question, we will review the terms coefficient and reciprocal. I will ask students to think to themselves the meaning of the two terms. I will then call on some students for an answer. The definition of coefficient is given in the resource.
This is a direct instruction lesson designed specifically to strengthen students' ability to solve equations and inequalities with fraction coefficients. Fluency of this sort is how I partially interpret mathematical practice 8 (MP8). I will give little to no attention on the meaning of solutions (though the distinction between the solution of an inequality and an equation is an important concept for the students to understand).
The method for solving the problems in the lesson are given in two steps:
1) Undo any addition or subtraction
2) Multiply both sides by the reciprocal of the coefficient
I will then present an example. What I write I will expect the students to write. Each example comes with a check for understanding problem.
Students may now work in pairs or trios on this set of 6 problems. The first 4 problems can be solved in one-step, the last 2 problems in two steps.
Though I said I would spend little time discussing the meaning of solutions, I may discuss it a bit with GP4 - an inequality with a negative coefficient. We may evaluate potential solutions to the inequality by substituting in simple values for d. Solutions like + 2, +4, +6, etc. This serves as reminder to students of why the sign must be flipped.
The last two problems may require close attention as they require two-steps. You may want to stop students after the 4th problem in order to go over a two-step equation or inequality. I have seen that most of my students understand the order of inverse operations to solve equations and inequalities so I am not expecting this to cause any difficulties.
Students will now work on 8 problems independently. There are 4 one-step problems and 4 two-step problems. If students are stuck, I will guide them back to the steps given in the notes. As students are working, I will be on the lookout for proper use of the reciprocal. If I see an error, I may stop and ask the students a few questions: 1) What is the coefficient in the problem? ; 2) What is the reciprocal of the coefficient? 3) How do we solve an equation or inequality like this?
Before beginning the exit ticket, we will review the two-steps for solving equations and inequalities with fraction coefficients. Every student should know them by now! It might be worth cold calling students for a minute or so with a question: What is the first step? What is second step?
There are 5 questions on the exit ticket. They match the difficulty of the problems solved in the lesson. A successful exit ticket will have at least 4 correct answers.