At the start of today's lesson I put the following Bellringer work on the board:
Solve the following Equations for the value of unknown variable, x
1. 3(2x – 5) = 19
2. 18 – 7x = 62
3. 48 = 3/7x + 14
I allow my students time to work in cooperative groups to solve these equations. As they work I walk about the room. I am both assessing students and offering feedback that moves learning forward.
I chose the three equations above to achieve the following ends:
After my students spend time working together on these equations I will choose three students to model their work on the board. I plan to choose students whose work on paper (or their explanations to their peers) make reference to one or more of the points listed above.
Teacher's Note: See Activating students as owners of their own learning for more information about my on why I take this approach in my classroom.
In Todays Trajectory I talk about where today's lesson is headed. If you want to learn more about what I hope students will learn today, and share when they model their solutions to the Bellringer problems, take a look at this video resource.
As students take a final look at the Bellringer problems on the board I will pass out the Keeping Your Identity Activity. To begin, I ask students to take two minutes to read the narrative and answer the Questions 2 & 3 about the mathematical term, equality. I ask that they work on their own at first, quietly thinking and recording their own ideas.
After two minutes, I will ask partners to discuss their answers for another two minutes. I encourage students to take notes as they hear new or different ideas from their partners. When the conversations ebb or veer off course, I will pull the class together to discuss today's learning target. I'll say something like:
Today we are going to learn why we use certain operations to solve equations. Our goal is to understand how using these specific properties enables us to solve many different types of equations successfully.
Teacher's Note: Watch Clarifying and Sharing Learning Intentions to learn more about why I introduce learning targets in this way.
Following this entree into the conversation, I will ask students to share answers to the equality questions on the handout. As they share I will scribe their responses on the board. I want to give as many students as possible to say something like "equality means both expressions have the same value" in their own words and see it on the board.
Next I will give my students five minutes to work in groups answering "yes" or "no" to the given problems in Part 4 of the Activity. As they work, I move about the room observing and asking probing questions. I am listening for students who are using terms or language from our opening discussion. I am observing who is looking at the board reflectively (or with a puzzled look).
After five minutes or so we will complete a Mini-Wrap Up. I have already assessed where many students are with respect to these problems, but we'll go over the answers and I will answer students questions. If necessary, I will use one probing question to maintain the momentum with respect to explaining equivalence. If students give me a one word Yes-or-No answer I will respond by asking their group, “How did you prove or disprove this answer?” I want students to consider this an important step in the process of working as a group: using proof to validate answers.
During this section of the lesson I want my students to be actively engaged in group work so that they are a resource for one another. They will start by working through the next section of Keeping Your Identity. I'll give students about ten minutes to get as far as they can, hopefully completing the section where an input is needed to form an equivalent equation. Some of these questions are a little tricky. I am targeting particular things that my students often struggle with because they don't drill deeper to understand the structure of an equation (MP7). We will use their work together in this section to begin a discussion on the number of possible solutions for an algebraic equation (i.e., some equations have one solution, some have two solutions, some have no solutions, and others have infinite solutions). The number of solutions will be a key focus of our Mini Wrap-Up at the end of this section. To prepare I am listening carefully for students to call on as they work on these problems.
When my students get to Question 5, I am looking for them to recognize the need for the Identity Property of Addition: the only value that you can add to a number that does not change the identity of the number is zero. Similarly, I want them to recognize that the number 1 is the only way to have a multiplicative identity. At the same time I am looking out for students who are "cancelling". Too many of my students learn to “cancel” numbers in steps as they solve equations. The sequence of tasks in this lesson helps my students build a deeper understanding as to why certain operations are extremely helpful when when solving an equation because they reduce an expression to a multiplicative identity (1*x) or an additive identity (x + 0). As students to simplify expressions by writing them in equivalent forms, I am observing their work habits to make sure that they are attending to precision (MP6).
When I see that most students have explored Question 5, I will bring the class together for a few minutes to discuss what they have learned. Then, we will work together to complete Question 6. My goal is to give students time to think about what they have done so far, then prepare them for working independently on Questions 7-9. I try to pace the lesson so that students have about 10 minutes to work through these questions. We may not have time to discuss these questions during this class period. This does not require a change in the homework assignment (Question 10). The discussion of Questions 7-10 is an appropriate way to begin tomorrow's lesson.
For homework tonight I assign the practice problems in Question 10 for homework. I encourage students to discuss any problems they have using Edmodo as they are work from home. Depending on how today's lesson went, I may post some fully completed answers to some of the more challenging tasks that we worked in class today.