##
* *Reflection: Checks for Understanding
Variables on both sides?... No Problem - Section 2: New Info

Students will always do the following when subtracting a quantity from both sides of an equation. (similarly when performing any other operation on both sides)

3x + 6 = -5x - 12 -3x -3x

In this case the student subtracted 3x from both sides, but notice that the student placed the expression being subtracted directly under the 3x and the 5x. This is fine. But does the student understand that he/she is subtracting 3x from the entire left side and the entire right side?

I find too many students do this disconnected from the the concepts of equivalence and combination of like terms. . Each time I see this I suspect the latter. I ask questions like "why not place the -3x under the 6 and the 12?" The student should respond that it doesn't matter because one will end up combining like terms anyhow. Too many times I find that students become confused with this question. I then make them see that they are actually subtracting from the entire side and asking them to simplify 3x + 6 - 3x, so that can see that the result on the left side is 6.

This misconception is clearly visible when students decide to divide by a value on both sides and do as follows: given 12x = 30 - 3x

12x = 30 - 3x 3 3

*Subtracting from both sides. Search for the misconception.*

*Checks for Understanding: Subtracting from both sides. Search for the misconception.*

# Variables on both sides?... No Problem

Lesson 6 of 10

## 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.

*60 minutes*

#### Launch

*10 min*

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.

#### Resources

*expand content*

#### New Info

*15 min*

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:

- Subtracting 5x from both sides
- Adding 4 to both sides
- 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.

#### Resources

*expand content*

#### Application

*25 min*

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**).

*expand content*

#### Closure

*10 min*

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.

*expand content*

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- UNIT 1: Number Sense
- UNIT 2: Solving Linear Equations
- UNIT 3: Relationships between Quantities/Reasoning with Equations
- UNIT 4: Powers and Exponents
- UNIT 5: Congruence and Similarity
- UNIT 6: Systems of Linear Equations
- UNIT 7: Functions
- UNIT 8: Advanced Equations and Functions
- UNIT 9: The Pythagorean Theorem
- UNIT 10: Volumes of Cylinders, Cones, and Spheres
- UNIT 11: Bivariate Data

- LESSON 1: Solving Equations with Tables and Graphs
- LESSON 2: Telling the story of x
- LESSON 3: Solving Two Step Equations
- LESSON 4: Clean up before solving
- LESSON 5: Fractional Coefficients are no problem
- LESSON 6: Variables on both sides?... No Problem
- LESSON 7: What's the problem?
- LESSON 8: Situations that sometimes, always, or never happen.
- LESSON 9: Solving Formulas
- LESSON 10: Hybrid Cars