Variables on both sides?... No Problem

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Objective

SWBAT solve equations with variables on both sides.

Big Idea

Equations may look difficult, but once students see and know what to do, they become confident, and actually enjoy solving these.

Launch

10 minutes

I launch this lesson with a problem slightly more challenging than previous work students have done. The Entrance Ticket this time involves two problems, one with a fractional coefficient requiring students to use the distributive property. The other problem involves multiple fractions. It only looks complicated. Once started, students soon realize it's really no big deal.

As usual, I walk through watching students perform these two problems asking each to check their answers, before calling a couple of students up to the board to work out the problems and discuss any doubts or questions any student may have.

 

New Info

15 minutes

After the launch problems are discussed, I show the NewInfo -- Multi-step equations Powerpoint slides. As I project a slide, I will call on a volunteer to read it. I want my students to first understand that there are a number of ways to solve equations with variables on both sides. The difference between an equation like the one in Slide 1 and those that they worked in previous lessons is typically a single step.

  • Slides 3 and 4 show two very common errors students make when first confronted with these equations.
  • In Slide 6 that I braced the entire left and right sides of the equation and subtracted 3x from both sides. Many students simply write the -3x under the 3x On the lefts side, and under the 5x on the right side. This may or may not indicate a misconception, so I always ask questions to make sure.  (See reflection)

     Example:  3x - 5 = 5x - 23                                                                                                                                                                                                -3x        -3x

In order to help students become more flexible with their problem solving, we will discuss different approaches to the problems. I will ask my students to identify different, convenient first steps to solving the given equation. For example, students may suggest:

  1. Subtracting 5x from both sides
  2. Adding 4 to both sides
  3. Adding 23 to both sides

After we consider the merits of each starting point, I will ask the class to choose one of any of the possible routes and solve for x. Once they are done, I randomly ask for the solutions to demonstrate that the same solution is obtained,  no matter what route is taken. 

Application

25 minutes

The activity in today's application section, Equations with Variables on both sides, works well as a partner activity. (Each student should have their own worksheet.) The seven problems are in increasing level of difficulty, but I expect that my students will manage first 5 problems well. Problems 6 and 7 are word problems. Some of my students will have difficulty writing an equation representing the situation.  

For Question 1c, I will look to see how my students work. There are two possible answer for question 1c which apply to the idea that  a ÷ b = a · 1/b. I may decide that this is a good time to review this property with my  students. 

Question 4 asks students to produce two different solutions. Depending on how the lesson began, I may ask my students to divide up the work. One students can solve the equation using one path, the other can try a different routes. After, students compare answers and check each other's work (MP3, MP6). 

For Question 6, a common error is for students to think that the value of x is the solutions to the problem. If this happens, I simply ask..."ok,  Is this the question answered?" Most students will think for a minute and then realize that they must substitute x to find the perimeter, the result asked for in the problem. 

When I see students struggling with Question 7, I will consider asking students what they can do to clear the equation of decimals. If necessary, I will lead students to see that they could multiply by 10 on both sides to rewrite the problem with integer coefficients.

For today, I don't expect everyone to finish this worksheet successfully. Yet, those who work hard to do so, will have an easier start in the next lesson (MP1).  



Closure

10 minutes

When students finish the Application Worksheet, I will call on students to share their answers one at a time. After a student presents, I will ask students if there are any questions.

For Questions 4 and 5, I will ask the student answering to state the steps they used when solving. I will also ask the class if anyone solved the equations differently and to state what steps they took. 

I expect most students in the class will make good progress with Problem 6. Tonight's Homework will provide additional practice with these types of problems. We will also work on more word problems in upcoming lessons.