Inequalities with Negative Coefficients

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Objective

SWBAT solve two step inequalities with negative coefficients.

Big Idea

Students discuss and explore solutions to inequalities with negative coefficients

Do Now

10 minutes

Students enter silently, take a seat, and begin completing their Do Now assignment, which is already waiting for them. It consists of 4 inequalities that must be solved and graphed. Thus far, I have tried not to include any inequalities with negative coefficients until I introduce the topic today. I am focusing on remediating any issues with solving using opposite operations, distributions (question #4), as well as checking for understanding of graphing inclusive and non-inclusive inequality statements (closed circle vs. open circle). My data has shown that most students correctly interpret and graph the arrows in appropriate directions, but a few are not as consistent about the circles. I take these few students in each class and work with them during the do now in a booth. When these students arrive at their seats they see an invitation, not a Do Now assignment, directing them to a booth to work with me. The invitation reads:

Hello (student name)! Today you get to work with me on these types of problems. Please leave your backpack at your seat, and bring a pencil and a positive learning attitude to the booth where I am sitting”.

I review #1 and #2 with my small group by asking first, “what do I need to do to solve?” or “how do I isolate the variable?” If a student gets stuck, I respond with another question, “how do I get the letter alone?” or “what do we usually do with equations that look like this to get the letter alone”? It is important at this time of year to push students to recall important vocabulary or processes learned in algebra thus far. Students who can persevere in recalling this information are using MP1. Each time we get to the graphing stage I ask students to tell me what type of circle I should draw (open or closed) and why, as well as what direction the arrow should be drawn, and why. Students answering these questions for each other and for me are using MP3. I stop every other time I ask a question to also ask the other students if they agree. Then I leave them to work together on questions #3 and #4. By this time, I have been working quietly with a small group and it is expected that all other students work silently on their assignment as well. After working with my small group, I will announce to the class that they can review the answers or complete the work in partners or small groups.

I complete solutions and graph them at the SMARTboard for one final check as students are allowed to ask and answer questions about the work. Then, we transition to class notes as I distribute this sheet to students.

Class Notes

15 minutes

Students are asked to fill out the heading and to copy the aim off the board into their class notes. They are asked to take 3 minutes to silently answer the five “True or False” questions at the beginning of their notes. The opposite side of the page is reserved for students to write an inequality to represent each statement. The answers are indicated in bold here:

1)      5 is less than 9                                                           5 < 9                       True

 

2)      10 is less than –12                                                    10 < –12               False

 

3)      The opposite of 3 is less than 2                          –3 < 2                    True

 

4)      The opposite of 6 is less than –7                        –6 < –7                 False

 

5)      The opposite of 1 is greater than –5                  –1 > –5                            True

 

We read the following note/question together:

Is the opposite of any number greater than 5?

I also ask students to copy my algebraic translation off the board:

x > –5

I explain to students that this algebraic statement asks the same question in the notes, “is the opposite of any number greater than 5”? Then I ask students to consider the question, “What does the number have to be in order for this inequality to remain true”? Or, what should x be for the statement to be true? I ask students to discuss the answer to this question with their neighbors. I warn them to think about all types of numbers, fractions, decimals, and negatives. I explain that using diverse numerical examples for x will help them construct an argument for the answer to a question. Students are in use of MP3 during this time. They are given 4 – 5 minutes to construct their answers and arguments. I use a graphic organizer as scaffolding/differentiation for students who struggle with the academic expectations of this task:

After discussion, 1 or 2 students/groups will be asked to share in their discussion, preferably if they have two distinct answers. We will review examples used to arrive at the answer, “the number must be smaller than 5” for the inequality to be true. The information written below will be revealed on a pre-made piece of chart paper and students will be asked to copy it into their notes.

expression a:                      If “the opposite of a number is greater than –5”,             à           –x > –5                

expression b :                   then “the number must be less than 5”                                 à           x < 5                      

Students will then be asked to note what happened to the sign from expression a to expression b, and 1 – 2 students will be asked to share out their thoughts, with my guiding questions such as, “why do you think that happened?”, “if this were an equation, how would be isolate a negative x?”, “what does this mean about dividing by a negative?”, and “do you think this applies for multiplication as well?”.

It is important to note that this is a detailed discussion where students may lose steam, but it is important for student to analyze the reasons why the sign flips so that small errors are not made. Whenever class lagged I would allow students to have more discussion time and I would float around the room to answer individual questions. I also attached the link below to the bottom of students’ papers and encouraged them to view the video at home if they were still confused. In my one hour class I needed to keep this section to 15 minutes or less, but in a longer class I would not hesitate to make this a 20 minute section to truly discuss the depth of understanding of the topic.

Task

20 minutes

I make two tables available at the front of the room to work with a small group of students (no more than 6) who choose to work with me. Booth seats are raffled through a random name generator and students are given the opportunity to bring two friends over to work with them. Fifteen minutes are set on the SMARTBoard clock to help students keep time. They are responsible for answering 8 problems during this time. We take the last 5 minutes to review.

I order the questions by level of understanding from solving and graphing at the basic to application to word problems at the end (7.EE.A.1 – 7.EE.B.4) Even if students struggle with only the first four problems, they are practicing targeted standards important to their grade. I use white boards with the students choosing to work with me. Each time, after solving our problem on the white board, I ask students to copy down the work on their papers. If we are running out of time I make sure to review one question from each of the three sections in the worksheet. 

Closing

10 minutes

For our closing, I show students all answers and they take 5 minutes to check their work and complete a journal entry copying and answering the following prompts:

  • What I know about solving and graphing inequalities with negative coefficients so far is ____________________________________________________________________ .
  • What I'm still not sure about is __________________________________________.
  • I would like to see more examples like problem(s) # __________________from the task.

I collect journals as students exit the class. I particularly like this journal entry because I use the problems from the last question to build homework, do nows, and thinking skills (morning work) for students.