As will be the case every day this week, today's opener consists of a series of patterns problems. Just like yesterday, today's problems review what we've done so far while adding a few new tidbits of mathematical knowledge. That newness begins with the framing of the problems:
The rule for a pattern is t(n) = 3n - 22.
That's how I quietly introduce function notation. I go about my business, circulating throughout the room and getting everyone started on the day's work until the first student asks, "What's t(n)?"
I respond, "Thanks for asking about that - now I know you're paying attention!" I ask how many people have seen this notation before, and it's brand new to virtually everyone. I say that this is called "function notation," that it's a very powerful tool, that you'll see it a lot in your continuing study of mathematics, and that "I feel excited and honored that I get to be the guy who gets to show you this for the first time."
The first question is about the first five terms in the sequence. Up to now, our "rules" have just consisted of the right-hand expression side of things, and students have grown comfortable using these. It doesn't take too long for students to tell me that the first few terms are -19, -16, -13, etc. I simply build on this knowledge by writing these values on the board using the new notation. "The first term is -19," I say, writing:
t(1) = 3(1) - 22 = -19
I ask the class: "Can you see how I indicated that I'm talking about the first term?" I want kids to see function notation as a tool (MP5) that arises naturally from a need for efficiency in mathematics. When they see it this way for the first time, especially after having looked at pattern rules for weeks now, this feels sensible to most students. "So how would I talk about the second term?" I ask. We continue, and I line up the first five terms. It really clicks for a lot kids when, beneath these initial five terms, I add the 200th term.
This gentle introduction really seems to work for a lot of kids. I still have to be careful to help the students who read "t(n)" as "t times n", but even there, I tell them that I'm glad their paying attention (and we talk about homonyms, homophones, and homographs). When we read the notation aloud then, it really does feel reasonable and natural to say, "Term 1 is, Term 2 is, Term 200 is..." I save the words "t of n" for another day, again in an effort to make this make sense.
The third opening problem was new yesterday: given the value of a number in the pattern, can you figure out what term it is? Kids are still making sense of this, but now I can write
t(n) = 281
before asking what n will have to be and writing the equation that we'll solve.
The fourth problem introduces yet another new idea - inequalities - and with some classes I won't get to this today. But I allow students to grapple with it, and if there's time, we'll run through the example. Here's your hint, however:
t(n) >= 0
Distribute Mastery-Based Progress Reports
Today is the first time that students are seeing their mastery-based progress reports for this class. I distribute reports to each student, and help them understand what they're looking at. There are three levels of organization here. First, learning targets are split into Mathematical Practices and Content categories. In each category, the learning targets are written in order, then beneath each learning target at the assessments and grades for each.
I share this example with students: Example Progress Report 1, and I show them the levels of organization. I remind them that Mathematical Practices are graded as an average of all scores, while the content targets are graded by Maximum Value.
We also note that everyone has a "U" on SLT 1.1, which is our focus this week, and that the three quizzes listed for this SLT haven't happened yet - they're dated for the next three days, and the "M" grades for each are placeholders.
Return Number Line Projects
In order to help students even better understand my implementation of mastery-based grading, I ask them how many grades they can find for the Number Line Project. There are four: beneath MP1, MP7, SLT 1.02 and SLT 1.03. For the majority of students, the grades are different between the learning targets. Once we've noted this, I return graded projects to everyone. I tell students to look at their project rubrics for how they've been graded, and then to find the corresponding grades on their progress reports.
"What you should see here is that I don't just give you one big grade for the whole project," I explain. "You receive high mastery scores for what you've done well, and you get low grades for what's incomplete or not quite right. This means that you can give yourself a pat on the back for your good work, and you can make a plan for how you're going to improve wherever improvement is needed."
I explain that all the parts of the Number Line Project are great tools, and that students will want them for reference as the year moves on. I give students a few minutes to make sure that their binders are organized and they've filed their projects.
I show students the frequency distribution of score ranges in their class. Plenty of students are below that 1.6 threshold right now, and few are above 3.0. It's important for them to see this, and for us to think about how we're going to improve upon that. Again, we're building a growth mindset here. "What matters is what you've learned by the end of this class," I say. "Your grade right here is not a final grade - that will come at the end each marking period and at the end of the year. Let's continue to work hard, learn as much as you can, and that's how you'll be successful in this class."
Every day this week, students will spend time solving linear equations. One important element of this work is that I've defined eight levels of equation solving for students, and it's up to students to try to progress through these levels. Today, I provide another Kuta Software worksheet that has four equations at each level from 1 to 6. There are only 10-15 minutes left in class by the time we get to this, but that's enough for many students to race through the first few levels. Tomorrow, students will have the chance to choose the level of the quiz they'd like to try.
My two most important talking points with kids are as follows:
As indicated on the weekly homework sheet that I distributed yesterday, tonight's homework is a set of one-step equations in the textbook.