Today's opener (on the first slide of today's lesson notes) helps make a bridge between the just-concluded systems unit and the just-starting unit on quadratic functions.
The first number riddle looks familiar to everyone. Students are comfortable finding a pair of numbers whose sum is 48 and whose difference is 16. Some will ask, "Didn't we already do this?" to which I'll wink and commend them on their attention to detail before saying, "What about the second problem?"
It's the second problem that's new. Some students will need a reminder that the product is what you get when you multiply two numbers. Some will jump right into writing the system
x * y = 48
x + y = 16
and they'll ask how to solve this. Others will wonder how to get started.
To everyone who hasn't thought of it on their own, I suggest using guess and check (tomorrow, we'll see what that system looks like on graph). "Who can tell me two numbers that multiply to make 48?" Most kids start with 6 and 8. "So what's the sum of 6 and 8?" I ask. We see that it's not quite 16, and that we'll need to try again. Sometimes students will run with the problem from here, and other times I'll see that it might be better to shift gears and ask for pairs of numbers that sum to 16. Either way, we informally run through a list of a possibilities until we settle on 12 and 4.
The purpose of this problem, of course, is to prepare students to factor some quadratic expressions. The puzzle I'm serving up here is exactly the one students will have to solve when they want to factor the expression
x^2 + 16x + 48
and number sense will certainly come in handy here. I'm introducing this puzzle with the belief that it's inherently satisfying to find two numbers with a certain sum and a certain product. I've seen that the sooner kids can embrace this as a neat little puzzle they get to solve, the sooner they will master factoring quadratic trinomials.
The purpose of the rest of today's lesson is give students a chance to play with that compelling task - and if they don't already find it compelling, I'm hoping that today's lesson with help to build their curiosity.
Today is the first day of the new unit about Quadratic Functions, and it's also the start of final marking period. I put the words "TEN WEEKS" on the board in big block letters, and I tell the class how impressive it's been to see all that everyone has accomplished so far this year. Then I make my pitch: "I want all of you to do your best work of the year over these next ten weeks. If you choose to finish strong, you're going to be very proud of what you've accomplished."
I give everyone a moment to think about that as I return work: I have Unit 5 exams from last week, and other work that students turned in during their mastery sessions. Students file what they receive in their class folders, ask questions about grades, and compare notes with each other.
As students size up what has a become a substantial body of work in each of their work folders, I ask if it feels like the year has gone fast or slow. I hear a mixed chorus of both as students consider the question. "Well, let's get to it, then!" I conclude.
Review of Area and Perimeter
Today's classwork, Building a Rectangular Garden, will reframe the sum and product puzzle from the opener in the context of rectangles with given area and perimeter. This assignment also makes a soft connection to some interdisciplinary garden-building work that's newly underway at my school. (That project is going to develop over the next year or so. You'll probably see parts of it beginning to pop up in this curriculum!)
We get started with some scaffolding. At the top of the page is some space for students to write the formulas for area and perimeter of a rectangle. I provide the image of a garden with an overlaid grid to give a clear image of what area and perimeter look like on a rectangle.
I project the fourth slide of the lesson notes on the board so students can see the variable definitions and the formulas. We spend a moment comparing the two versions of the perimeter formula. I want students to note that they both accomplish the same thing. I say that it's optional to write both or to just choose one, and that I prefer (for reasons we'll get to) the second one.
I take just a moment to make sure everyone agrees on the area and perimeter of this first rectangle.
Introduce the Problem
Now we can move on to the task at hand. The perimeter of the first rectangle is 24 feet. Task #2 says that students should draw a new rectangle with the same perimeter and a different area.
I've left this task open-ended because I want conversations to start right away. What happens pretty naturally is that students start to wonder about how many possibilities there are. I invite students to draw their rectangles on the board, and here are the results. It's great to begin by getting this visual of the different rectangles. There is so much to talk about here. We might discuss the nature of each rectangle: is it long and lean or short and stout? What does it mean for the area when the dimensions are close in measure? Farther apart? The same? Is a square a rectangle anyway? (To which I respond, "Is a puppy a dog?")
Some students will notice that it's got something in common with the opener. Others will want to make a list like this, of all the possibilities. I love it when students get to that point on their own, because that means they're getting hooked and are going to enjoy factoring. I steal that idea, making my own chart on the board, and suggesting that it's unnecessary to stick to just whole numbers. I also ask if there could be such a thing as a rectangle with length 12 and width 0, which leads to all sorts of fun, and which students will get to consider again tomorrow, as they look at a graph and consider extraneous solutions.
Task #3 instructs students to take whatever rectangle they made in #2, and to draw another rectangle with the same area but a different perimeter.
Here and on the back of the assignment, I'm asking students to draw each rectangle. There are several reasons for this move. I want this model to stick in kids' minds as we move on to more difficult problems. I want students to visualize how a rectangle with relatively large and small perimeter will look. I'm able to look over kids' shoulders and see who's drawing the grids and who doesn't need to (or who realizes that they can drop the grid, mid-assignment). I want students to play with ideas as they get to the more difficult (or even "impossible") rectangles to draw. It's one thing to draw a rectangle with perimeter 20 and area 21, but what about area 20? What about area 26?
Back: Rectangles with Perimeter of 20
The front of the assignment and the conversations that accompany it help everyone to understand this sort of problem. When students get to the back, which consists of a series of similar problems, almost all of them are able to jump right in.
As students work, I circulate to each table and ask everyone what they notice. There are no particular responses I'm looking for, and my main goal is to get some conversations going. I let students work right up to the bell, and I tell everyone to finish this tonight for homework.
Here's the Catch
The sweet spot, if I can time everything right, is for most students to be working through the back of this assignment - but not quite done - when the bell rings. Problems #10 and #11 are special cases here. Can you see why? The real point of this assignment isn't just for students to practice working with area and perimeter. I'd really love for kids to spend a little time tonight being stumped on each of these before we figure out what's going on tomorrow.