As students enter the classroom, they are prompted to list all the factors of 360 on a sheet of loose-leaf paper. Depending on the class, it’s always interesting to see the ways different students grapple with this problem. To be completely honest, I’ll also note that students occasionally take longer than I wish they would on what is meant to be a quick opener. It’s important not to be disheartened if this happens - rather, this means that it's precisely the challenge kids need.
This warmup is a robust opportunity to develop "math muscles," and it is immediately embedded in the task at hand. I think this is one interpretation of what the CCS mean by coherence. Most of today’s lesson involves a lot of abstract reasoning; here we begin by reasoning quantitatively (MP2). In addition to this being an exercise in number sense, there is a practical purpose here: we’re going to build the Big Chart on Excel, and in the first column, we’re going to need all of these numbers.
To make sure we move along, I avoid mystery and tell students that there are 24 Factors. As I walk around, I remind groups of students that the square root of a number splits its factor list in half. When I see that a student is making use of the fact that all factors come in pairs, I make sure they’re sharing that idea with other students. When I see that someone recalls our recent conversation about 360 being the least common multiple of the 9 out of the first 10 counting numbers (everything but 7), I make sure that everyone else at their table remembers this too. Students often get to 22 factors and need a nudge to find 24, 45, or 72 – when I see this, I again remind them to make sure that every factor has a partner, and this helps them make sense of the problem (MP1).
When students finish, they get a laptop and go to the class web site, where they follow the next steps in today’s agenda. Click on the link above to see the agenda on Wordpress.
Students follow the instructions on the agenda to download and rename the fileDPP Big Chart Blank. They enter their radius at the top of the sheet. They naturally try to move directly on to Circumference and Area, which in turn generates the need to know how to enter π in Excel; I tell students to look this up on the web before I tell the class how to do it a few minutes from now. I try to encourage such research moments as often as possible, especially when we have the computers out. If you’re reading this on Better Lesson right now, you already know that you can use the web to learn about anything you want. I think it’s important to establish this habit of self-sufficient scholarship in my students as well.
Excel Big Chart Intro gives an overview of the blank Excel file that students receive.
As students open up the file, I circulate and tell them to "enter all the factors of 360, from greatest to least, in the Central Angle column of the chart." In an effort to encourage precise use of language (MP6), I consistently use this wording in my first trip around the classroom. I then take another lap of the room, and invariably find students starting from 1 and going up. I remind them again that the list should go from greatest to least, and I ask them to tell me what that means. After a few minutes, the entire class is set up, and it’s time for today’s mini-lesson.
Mini Lesson: Writing Formulas in Algebra and in Excel
Teacher's Note: If you haven’t already, take a look at the second page of Part 4 of this project. There is a chart that requires students to write both the “Algebraic Expression” and the “Excel Formula” for each of the values they have been working to find since Part 2 of the project.
I begin by using Circumference and Area as an example of what this means. I draw this chart on the board, and I ask students for the algebraic expression we use to represent the radius of a circle. Everyone feels good when we decide to write “r” in that first box. Then I ask, “If r represents the radius of a circle, what expression represents the circumference of that circle?” Note that I’m not using the word formula here – I’m asking for the expression. What I’m looking for, of course, is 2πr: this expresses the circumference of the circle, in terms of r. We can write “C=” in front of this to make it a formula, and you can see I have done this, but I carefully note this to students: “C” serves a signifying variable we can use as shorthand to identify “circumference,” and writing it here allows us to reference it later. But to find “C,” we need the expression in terms of something we already know. We quickly repeat these steps for area.
This brings us to Excel formulas. I make an emphatic point that the Excel formula is going to be exactly the same thing as the algebraic expression, it’s just that we need to represent some parts of it a little differently. To illustrate this point, the first thing I do is point to the “2” in “2πr,” and say, “How do we say ‘2’ to a computer?” Students feel safe with this one, and I write the 2 in the Excel column. Now, in algebra, I explain, we know that if a number is next to a variable or π, with no space in between, that means we’re multiplying. In Excel, we need to explicitly indicate what we want to do. I ask for the symbol for that, and I write the asterisk after the 2.
Next we need π. There are two varieties of questions students might ask when it comes to using π in Excel. One is from students who recall that the TI-83 has a button for π, and they realize that Excel should have one too, so they ask how to find it. The other variety is some version of, “How many decimal places of π should I use when I type its numerical value into a formula?” The latter variety is more interesting to me, because this question indicates that students have a bit to learn about how using this tool allows them to attend to greater precision. When I show them that Excel has a function for π, I make the point that Excel stores very precise values. Even if it is displaying a number to just 2 decimal places (see video), it stores a much more precise value. I explain that we’re not going to any rounding ourselves: we’re leaving it all to the software in front of us.
I ask if anyone figured out how do that, and there are always students who have, and it’s usually funny to listen to them try to explain it. “You have to write pi,” they say, and I write the symbol on the board, “no, no, no, the letters…p…i…then there are these parentheses…” Good, I say – now what do we make of those parentheses? I ask the class to recall algebraic function notation, and the parentheses in “f(x)”. Excel functions will always have parentheses as well. I leave it at that for now – adding parameters inside those parentheses is part of tomorrow’s lesson.
Now that we have “2*pi()*”, we have to figure out how to represent the radius. I treat this as new information even though I know it’s review for many students. I point to my screen and I ask if anyone has played Battleship before, then I say this spreadsheet is the same idea. I can refer to any “cell” by naming its “column” and “row”. So then, where did I write the value for the radius? We see that it’s in B1, and I write that on the board. Finally, with my left hand I point to the algebra and I put my right hand on the Excel formula. As I move across both, I say, “2 times pi times r – hopefully you can see that both of these are really the same thing.”
Finally, I move back to my computer and type our formula into Excel. I dramatically press enter…and nothing changes. I ask if anyone can tell me what happened. Several students shout excitedly that we need an equal sign! Exactly – in order to make this a formula, we’re always going to need an equal sign, which tells the software “this cell is equal to the formula I’m typing here.” We repeat this process for area. When that’s done, I show students that now we’re building a dynamic spreadsheet. I change my radius value in B1 to a variety of different numbers, and we watch the circumference and area change. This is what the software will allow us to do.
Filling in the Big Chart
With circumference and area serving as examples, it’s on to the meat of Part 4. I show students that in chart on the back of Part 4, they will work through the same process of writing an algebraic expression and an Excel formula for each of the fields on the Big Chart. I ask them all to make sure that they have their first column filled in, with all factors of 360 from 360 down to 1. Now we have to figure out a formula that will take the central angle and give us the number of slices.
To elicit this formula, I ask students to look at what they wrote for their description of the relationship between central angle and number slices on Part 3 of the project, and for a volunteer to read. Look at this photo to see a student’s words on the board, followed by the equivalent algebraic expression. I demonstrate this process to help students move from quantitative to abstract reasoning. We all agree that this formula works, but now I point out that we already have the central angles written in our spreadsheet. We don’t need a formula that takes the number of slices and gives us the central angle – we need the inverse of that. Students take a moment to invert the formula, which you can see in the chart to the right in the same photo. Just as we preceded the expression “2πr” with the signifying variable “C=”, now we can say “n = 360/theta”. This will allow us to refer back to the variable n as we continue through the project.
Number of slices shows how I enter the formula for “# of slices” into Excel. Note that on the board, I have only written the formula I typed for its first instance, by using “A5”. I will continue to ask for the first instance of a formula throughout the project, and I’ll assume that students have mastered the click and drag for propagating the rest of the column.
Now that we have the number of slices, I ask students to name other parts of the construction sheets that were “easy” to fill out. They know that number of triangles and number of sides are the same as the number of slices. We just need to use same variable, n, and the Excel reference only needs “=B5”. For students, this is a reminder that equations don’t always require a calculation, they just need to say this is the same as that.
Work Time, with a Quick Note on Anchor Cells in Excel
With this complete, I set students to work on the project. They can work together or work alone to try to generalize the expressions and Excel formulas for each of the calculations they have been making throughout the project. This is our work for the rest of today and all of tomorrow’s class. I encourage the use of informal expressions, and I encourage students to try things on Excel: it’s ok to be wrong often as they make moves on their spreadsheet – that’s part of perseverance in problem solving. They’re into it. The Excel environment is highly responsive to testing out an idea, with feedback that is instant, unforgiving, and impersonal. You can’t get mad at a computer for not doing what you wanted it to do – you just have to figure how to say it right!
For the most part, I really want to give students as much space as possible to grapple with this task. There’s one more quick point I end up making a short time into the work period, however, and I illustrate it in a video, arc length needs anchor.
Students will have other questions about the remaining columns, and I tell them that I’ll help out tomorrow. For now, they should try things: write the algebraic expressions, test conjectures in the spreadsheet, search the web for necessary Excel functions. We’re building some capacity and creating a need for tomorrow’s lesson.
It's important to give students time to save their work where they can get back to it tomorrow. At my school, there's no guarantee they'll have the same computer tomorrow, so I make sure their using Dropbox, Google Drive, or their email to save their Excel file remotely. With about 10 minutes left in class, I tell them to that, to put computers away, and to complete today's record sheet prompts.
Record Sheet Prompts: