Two Step Equations in the Form px + q = r

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Objective

SWBAT solve two-step equations in the form px + q = r by working in pairs.

Big Idea

Students work in pairs to solve and check equations in the form px + q = r using algebra text books to check their answers.

Do Now + HW Check

15 minutes

Students enter silently according to the Daily Entrance Routine. There are Do Now assignments at their desks. Exit ticket data shows that students are struggling with negative numbers in equations we’ve reviewed this week. Thus, a majority of the problems and work included in the Do Now address negatives. In the first equation, x – 6 = –18 the opposite operation is add 6 on both sides. Some students falsely add 18 on both sides. It is important to reinforce that we want to isolate the variable, and –18 is on the opposite side of the equation. 

The next question serves as a check into positive and negative decimal operations. The answer to this equation is negative. An integer operations rule can be used to solve this problem with rational numbers because they follow the same rules.

In the expression x – y, if x < y then the answer is negative. Find the difference of the absolute values.

It is important to continue reinforcing the rules to motivate students to look for and make use of the structure of the problems (MP7). This in turn also reinforces the conceptual understanding of the combination of integers and rational numbers. For question 3 I reinforce the step of dividing by negative one to solve for positive t. If time permits, I illustrate two different solutions:

–15 – t = 7
+15        +15
–t = 22
–t/–1 = 22/–1
t = –22

or

–15 – t = 7
        +t       +t
–15 = 7 + t
–7    –7    
–22 = t

 

Problems 4 and 5 review previous lessons and use negative fractions and integers. Equations are written in different formats to get students comfortable with the diversity and to encourage analyzing the operations between the variables constants and among the variable terms.

HW Check

I use a random name generator on the SMARTboard to have students call out answers to the homework. Answers must be called out in the form “x equals _____”. I make sure to ask for questions after each answer has been called out. Students may ask to see the problem solved or explained as some of the answer may involve negative number operations. This is a good time to spiral back to the rules reviewed in the first unit. After all answers have been called out, I ask for a student volunteer to read the answer to the riddle. Some students may need clarification on how to find the riddle by scratching out each answer and readying the remaining letters.  Homework is organized and tasks are distributed.

Class Notes + Guided Practice

15 minutes

Students are asked to copy the aim which is written on the white board at the front of the class. Then, they are asked to locate the notes from the previous day and copy the steps for solving a two-step equation.

Step 1: Isolate the variable term.

Step 2: Isolate the variable.

From time to time I do ask student to re-copy notes from previous days to reinforce the use and habit of note-taking. This is a skill that many of our former students have shared is a skill they wish we had helped them develop for high school. 

I begin reviewing the solutions for these questions by modeling my thinking and my use of the steps. I let students know, before I begin, that I am modeling and that they should go through the same line of questioning and thoughts that I am using. The following narrative illustrates the procedure for #1, but the same line of questioning is used for each subsequent equation:

  • What’s the first step?
  • Isolate the variable term.
  • What is the variable term?
  • 5x
  • What’s in the way? (or) Why is it not isolated? (or) What else is there on this side of the equation?
  • Plus 16  (I have students say the sign in front of 16, not just the number 16. I feel that it reinforces the opposite operation)
  • What’s the opposite of “plus 16”?
  • Minus 16
  • Where?
  • On both sides!
  • Now what’s the next step?
  • Isolate the variable
  • What’s in the way? (or) What operation and what number are not leaving x alone?
  • Times 5
  • What is the opposite of “times 5”?
  • Divide by 5
  • Where?
  • On both sides!

I call for choral participation on about half of these questions. Then, on the next equation, I run through the same series of questions and cold call different students to answer each one. As I continue to guide students through the remaining questions, I may call on individual students to run through the questions AND the answers listed above. For example, 2 – 3 students may be responsible for asking and answering the questions for one equation. Once we get to the end, students work with neighbors to complete the last two questions. If there is still time left, students may ask clarifying questions. 

Task

20 minutes

At the end of 15 minutes given for guided practice students are asked to take out a blank sheet of lined paper and write a heading at the top. They are also asked to copy the aim on the paper and title it “Task”. Then, I project directions on the SMARTboard that read:

 

  • Turn to page 127 and complete problems 1 - 10
  • Write the EVEN answers only on the scrap paper. Be sure to include your name. This is your HW PASS opportunity.
  • Check your ODD answers on page 761. Look for exercise 3 - 3.

 

Big Ideas has several student dynamic editions available on their website. If you lick on that link, select chapter 3, and scroll down to page 112, you will find problems similar to the ones given during this class. For example:

 

  1. 2x + 4 = 14
  2. (1/2)x - 9 = - 5
  3. - x/5 + 3 = - 1

 

All students will have 15 minutes to complete the ten problems. They will be given a small sheet of recycled paper for their even answers mentioned above. This will serve as motivation for them to complete these questions and also will serve as an exit ticket for me to use as a check for understanding for the class. Students who get all problems correct will receive a homework pass. Students who either do not turn in a slip or get more than 2 wrong will be pulled for re-teaching and review during “Remediation” for each class. As I walk around the classroom to ensure students are following the correct steps and checking their work, I will be encouraging them to check their odd answers in the back of the textbook. 

Closing

10 minutes

Students will turn in their slips with even answers. All work and answers should remain out on their desks on the lined paper they were instructed to use. I will read out the even answers and ask students to raise their hand if they need to review the solutions for any of these problems. The rule of thumb I ask them to consider is, “if you know where you made your mistake, that’s great. But if you don’t know why you got a problem wrong and you can’t find your mistake, you should ask me to show you the solution”. Once questions have been answered or once we have run out of time homework will be distributed and class will be dismissed.