Note: Much of today's lesson is in response to the work I saw students complete on the Number Line Problem Set, which I collected during the previous class (on a Friday), graded over the weekend, and will return today.
For today's opener, I post four partially completed number lines on the board. Each number line has a beginning number, and end number, and either four or five spaces in between. You can see them here: Openers Sept16.
As students enter the class today, I tell them to complete all of these in their notes. Here's how it usually shakes out: students find the first one pretty easy, then they begin to have questions. I encourage them to try some things on their own and to talk to each other.
While students try these problems, I return their graded Number Trick Projects and Number Line Problem Sets. This allows me to move around the room, and to continue to make sure they're on task.
Solutions as Today's Notes / Mini-Lesson
When I'm done returning work, I go through a mini-lesson on how to fill in the blanks. I tell the class that the first part of their new project will be just like this. Then I say that there are two steps I usually follow when filling out these number lines. First, I think about how far the starting number is from the ending number. In the first example, the distance from 0 to 20 is 20. I write this number on the board, just to the right of the problem. (You can see my notes here: U1L13 Opener.) Then, I count the number of spaces in between the first and last number. In the first example, there are four spaces. I complete the division problem 20/4, whose quotient (5), is the number we'll count by. On the first problem, it all moves much faster than I'm describing it here.
We proceed to use this idea on the next few examples. "How far is -1 from 1," I say, setting up the fraction 2/4, then reducing it to both 1/2 and 0.5. Here, I make a strong point of saying that I'd like students to use both fractions and decimals whenever they can. For this example, I write the fractions above the line and the decimal values below. On an upcoming part of the Number Line Project, students will place fractions on a number line, and I'll ask them to write the values as fractions and decimals.
At some point while I'm doing this, I make the informal observation that these are similar to the pattern problems that students were looking at last week. We'll get deeper into that idea over the next few weeks.
You can see the other two openers worked out on this photograph: U1L13 Opener
To frame today's work, I stand next to today's agenda (Agenda) and I riff on the note at the top. "My goal is for everyone here to learn something in this class. That means I want everyone to improve from one project to the next. Today we begin a new project. If you didn't hand in the first project, your goal should be to hand this one in. If you turned in incomplete work, then your goal should be to follow instructions closely, and make sure you submit the right stuff. If you got a 2, then your goal is to get a 3, and if you've already done your best work you should be asking me for new and excellent challenges."
I note that I've returned all the Number Trick Projects and Number Line Problem Sets, and that students can see their grades on the project by taking a look at the project rubric. Then I say that I have a few statistics I'd like everyone to see.
I post the first graph: NTP vs NLPS, and as it pops up on the screen, I ask the question, "Do these 9th graders hand in their work?" I say that if the principal were to ask me this question, I'd show her this graph. Now: it's not pretty, yet. For the first project of the year, just under half of my Algebra students submitted their work. But I explain my optimism to students, and I strive to cultivate the growth mindset. On this graph, all signs point to our improvement. In every class, students can see that more students turned in the Number Line Problem Set than the Number Trick Project. What I tell the class is that to me, that means we're already improving. If we keep growing like this, we're going to be in great shape by the end of the year. "And now that you all know my expectations," I say, "you can work hard and do your best from the start of every project."
I show them the second graphic: Complete vs Inc 1, which shows the percentages of work that was complete when students in each class turned everything in. Again, these are not great numbers, but what I say to kids is that "it's September. If we finish the year like this, then of course this is bad news. But we're just getting started, and in a few months, we're going to be able to look back at this data and see how far we've come."
The Number Line Project begins today. In order to help my students begin to achieve this goal of improving throughout the year, there are more scaffolds and supports on this second project than there were on the first. As I mentioned two lessons ago in regards to the Number Line Problem Set, being well-acquainted with scale on a number line will help students produce better axes when we're graphing functions. By spending some intensive time on this over the next two weeks, students will practice with scale, exact and approximate values of rational and irrational numbers, patterns, and units of measurement. They will also practice staying as organized and as up to date on their work as I'd like them to able to be.
Week 4 Homework Sheet & Number Line Project Overview
In truth, this "project" really consists of a series of inter-connected handouts that students will work on, each at their own pace, over the next two weeks. To get started, I distribute this week's homework sheet, which is a little different from the others. It looks like this: HW Week 4
The top half of this page is a place where kids can write what they're going to work on each night. I say that homework is probably more important this week than it's been all year, but that each day in class, students will have to figure out what they're going to try to accomplish each night.
The bottom half of the page is a checklist of all parts of the project. In order to differentiate, but also to add a healthy dose of game design, I explain that I'm only going to give someone the next part of the project when I see that they're done with the previous part. When students finish each part, I'll take a look at it, help them with any major errors they may have, sign off on their checklist, and provide the next part. At the end of the project, students will collect all work, complete a cover sheet that includes a learning reflection, and hand in all their work.
I tell students that they have two weeks to finish this whole project, then I note that there are really ten parts to this project. Ten days of class and ten parts to the project mean that everyone should aim to complete one part each day.
Part 1 of the project is called "Numbers on the Number Line," as with the other two parts, it is divided into three sub-sections. This is Part 1a: Fill in the Blanks.
I distribute the handout while explaining that this is just like today's opener. It includes some pretty straightforward problems and some that are a little more difficult.
One of the key things that will happen on this part of the project is students will be able to think about fractions. Look at the handout. On at least #'s 2, 4, 8, 9, and 15, I'll want students to write the values of each number as both fractions and decimals. I show them how to write the fractions above the line and the equivalent decimals below, just as we did on one of the openers. On other problems, like #'s 10 and 11, I'll encourage the same.
For many students, this idea of using both fractions and decimals is annoying on #2 and #4, and I put up with a fair amount of complaining about the merits of decimals over fractions. What's amazing is watching students change their opinions on #8, which divides the line into sixths rather than quarters (#2) or fifths (#4). Once kids are confronted with the idea of writing 0.83333... or 5/6 they begin to come around on the idea that fractions may be a useful tool, and little argument is needed on my part.
Many of these problems are fairly intuitive, but most students find #13 and #14 difficult. In these cases, I offer help by making references to the algorithm that I introduced on the opener, and to the patterns we've looked at in previous days.
In keeping with the development of a growth mindset, I distribute todays exit ticket (Survey for Day1 of NLP) with a few minutes left in class. I certainly want to know what students will tell me on these surveys, but even more importantly, I think that the act of writing and filling these out is valuable. My students want to succeed, but many lack a language of success.
I recited a lot of these ideas at the beginning of class. Now students are thinking them for themselves. Part of what's challenging about developing a growth mindset is that many kids don't have the words for it. The act of completing this survey, in a tiny way, gives them the vocabulary to talk about improvement. Of course, I'll be curious to see what I get from them. But the real reason I give these to kids is for their experience.