You may want to use my expression or create your own.
It's important for many reasons that students are able to understand and apply the language of mathematics. They need it for basic mathematical communication, for all standardized tests including college placement exams, and to build their synapses (more vocabulary = better brain function!)
I begin this lesson with a complicated algebraic expression on board and listen to what my students say as they come into class. I anticipate that some will initially try to figure out how to "solve" for something while others will recognize that it's an expression, not an equation and will try to simplify it or identify what kind of expression it is. The challenge I pose for my students is to appropriately label all the components of the expression. This generates questions about what kinds of components can there be in an expression. Rather than answering those questions I have students work in pairs, using a textbook if necessary to complete the challenge. (MP1 and MP6) As students work, I walk around offering encouragement and redirection as necessary. I also keep an ear out for any particularly interesting discussions and ask those students to be ready to share what they've talked about with the class. For example, most students will identify (39/(x+4)) as a term of the expression, but only a few will recognize that taken by itself it is a rational expression with a constant numerator and a denominator of two terms (binomial). Another interesting discussion might be about whether to consider the third term as a negative term versus a positive term being subtracted. This is an example that allows for deeper thinking about what negative numbers are. I intentionally select each team to share some aspect of the labeling, so everyone participates. By making the choices as to who shares what, I can differentiate, allowing my less confident students to share the easier descriptions like the fact that there are exponents of 3 and 2. I might have a stronger team share that there are also exponents of 1 and -1. After everyone has finished, I ask the teams to share their discussions with the class.
You will need copies of the Scavenger Hunt, challenge, and Math Stumpers Game for this section. This section of the lesson will be in three parts: first students team up for scavenger hunt to find vocabulary, second, students individually complete a Challenge worksheet then self-check with the whole class, and third, students partner up to play a game.
Team work 15 minutes: I ask my students to work with their right shoulder partner to complete the vocabulary Scavenger Hunt. I might borrow a class set of Chromebooks so students can access the internet in addition to their textbooks for this activity. I remind them that they each need to copy the information and to be careful that the definitions and examples actually fit mathematically. (MP6) For example, they can find multiple definitions of the word "term" but most are general and not related to math. As they work, I walk around offering encouragement and redirecting as needed. Some students struggle with what I call "Mathlish" (a play on Spanish and English), usually because they don't really understand the words being used to describe the new vocabulary. I try to help these students make connections to things they do understand and to reword the definitions to make sense to themselves and/or to create examples to help better understand.
Individual work 10 minutes: When all teams have completed the scavenger hunt, I tellstudents that they will be working independently for the next activity. I hand out the challenge and ask them to read through the directions carefully then ask if there are any questions. I reiterate that not all the scavenger hunt vocabulary will be used for this activity but assure them that all the words/phrases will be used at some time during this unit. As they work I walk around offering encouragement and redirection as necessary. This is a great opportunity for formative assessment as I observe which students are struggling or using only certain kinds of answers. For example, if a student is only labeling coefficients and exponents, I might ask if they see any factors or terms. It might be that the student is just labeling the "easy" parts or it might be that they don't really understand the other components. (MP2 and MP6) When everyone is done, I have students self-check their work as I go through possible answers with the whole class. I read through the most common responses I've observed then ask for any additional answers. This self-check gives students immediate feedback about their own work and lets them hear alternatives in a non-threatening manner.
Game time 15 min: For the final part of this section, I let my students play the Math Stumpers Game using individual white boards (you can use blank paper instead). The game rules are simple and I find that my main duty during this time is to serve as arbiter when needed. (I only take that position as a last resort when two students cannot agree despite appropriate discussion) I've found that my students become more creative both in the problems they make and in their responses when they are in a game format. They actually push each other readily when it would take much more persuasion for me to get the same result using a traditional problem set! (MP1)
As we wrap up the lesson, I want my students to recognize the importance of using appropriate language when speaking or writing about mathematics. To help reinforce this, I give them a chance to "catch" me using the wrong vocabulary as I describe some of the things we've covered today. Examples of what this sounds like/looks like are in my Whatchamacallit video.