# Solving and Proving Equations with One Unknown

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## Objective

SWBAT...1. Solve one-step equations with one unknown. 2. Justify solutions to equations with one unknown.

#### Big Idea

Students think about solving equations as rearranging equations and finding alternative ways to "say" a variable.

## Entry Ticket

15 minutes

To open the lesson I give students problems where they have to solve equations with one variable. For the Entry Ticket: Solving and Proving Equations with One Unknown, I purposely give problems with addition, multiplication, division and multiplication with a fraction.

Here are four example problems:

1.  X + 4 = 11
2. 2X = 24
3. X/7 = 5
4. (3/4)X = 18

I allow students to struggle with these problems. I want to gauge the level of understanding that students have from the experiences in middle school.

When students complete the work, I ask them to compare answers with a peer. I encourage them to share their strategies and critique each other's work. I conclude the entry ticket by reviewing the answers as a group. I also stress that I am looking for students to prove their answer to the class indicating that checking the answer is a form of proof.

## Direct instruction: Active Note-taking

30 minutes

During this section I have students actively take 2-column notes as a way to learn and engage in the material. I have found that students take notes more efficiently and interact more with the material if it is written down as we go, rather than posted as on a projector.

Class Notes: Solving Equations with One Variable is included as a resource for teachers to use in this section. Teachers may also want to make copies to Differentiate Instruction for students. The notes are scaffolded by leaving more and more places for students to fill in information along the way.

In the notes I define the main terms for the lesson and then review a number of examples. I want students to identify the type of problem in the example in the left column and the step-by-step work in the right column. One example might be "Solving equations with addition - example: 4 + X = 11" in the main idea/example column, and then the steps to solve and check the solution goes in the right hand column.

To encourage active note-taking, I ask students to engage in short Turn and Talks. I also ask open ended questions so that students engage different domains of language (aka listening, reading, writing and speaking) during the process of gathering information.

To conclude this section I ask students to jot down a napkin statement where they have to summarize the notes with 1 or 2 main ideas - this gives students a chance to identify relevant details and encourages them to look at the examples and look for structure in expressions (MP7MP8).

## Paired Practice: Solving Equations with One Unknown

10 minutes

After the note taking, I have students work together on the following problems:

Prompt: Solve for X and prove your solution

1. X + 8 = 23
2. X -10 = 33
3. 6X = 84
4. X/11 = 121
5. (3/5)X = 33

The intent of this activity is to give students an immediate opportunity to practice the different types of problems reviewed in the notes. I encourage students to write down the work for these problems as a continuation of their notes because it provides more examples for them to refer to when they review.

In my class, our approach to this section of the class targets Mathematical Practice 3. Students are given the space and structure to not only identify their own arguments, but make sense of, and critique, the arguments of a peer.

## Independent Practice

20 minutes

Students are then asked to work independently on solving and proving solutions to equations with one variable. I will either assign a problem set from the textbook or use a worksheet. In terms of quantity, I tend to assign about 10 problems, trying to provide a continuum of difficulty. My goal at this point of the lesson is to give students a lot of practice with the skills of solving equations and verifying results. Kuta Software has a number of good free worksheets. Here is a free worksheet that I used in this lesson: Two-Step Equations Practice (source URL: Solving Equations in One Variable)

Once completed, I have students file their independent work along with the class notes and paired work in the class notes tab of their binder. From time to time, I will complete binder checks as a way to: (a) check on organization and (b) assess student work with varying levels of support. For instance the assessment might tell me that one student can do the problems with the support of the class and with a peer, but he/she is really struggling to complete the problems independently. With data such as this, I can successfully plan additional interventions for the student.

## Exit Ticket and Homework

15 minutes

For today's Exit Ticket I ask students to complete an Idea Organizer  that involves solving one equation, but pushes student thinking by having them show all of their work.

I like to focus on one problem for this exit ticket because I am want to take the time to emphasize the process and the Common Core emphasis on proving their solutions as a main part of the lesson. The intent of this lesson is not to get students to rote-memorize the patterns to solve equations but get to a point that they can prove and justify their work.

## Homework: Delta Math

For homework I assign a problem set on Delta Math. I typically have students complete ten problems in a row, with a penalty of one for an incorrect response. In other words, if students complete the first 8 problems correctly, but then get the 9th incorrect, then they have credit for 7 in a row and need to complete 3 more problems in a row to get full credit for the assignment.

For a great overview video on Delta Math and how to set up an account (it is free!), click here:Delta Math Overview Video