Students will investigate why some rectangles are easier to make than others. In doing so, they will develop an initial understanding of the nature of the roots of a quadratic function.

An un-solvable problem can be a great lesson hook!

20 minutes

Today's lesson notes are on Prezi, which makes it easy for me to "zoom around" the document that students will use today. In this video narrative, I give an overview of today's lesson and show an example of why Prezi is a great tool for a "guided notes" lesson like this.

**Opener, Part 1: Homework Collection**

One focus of today's lesson will be on some of the problems from yesterday's Rectangular Gardens assignment, which students finished last night. As class begins, I collect it just to see what students have got. As we dig into today's related work, I'll be more than willing to hand it right back to students if they say they need another look.

Of course, it's most fun to talk about the more difficult, and even impossible problems at the assignment, and these problems will take center stage later in the lesson.

**Opener, Part 2: Building Background Knowledge on Delta Math**

This lesson is designed to be used during a longer block class of 65 minutes. Having the extra time allows me to give kids a little while to build background knowledge before the lesson, but I've found that this initial practice is so useful that it's worth doing even in a shorter class.

Here's a screencast that shows what students have to work on now. As kids work, I circulate to check in, and I tell them how impressive it is to see that they're sticking with this challenge. "I know that it can be frustrating to race the clock like this," I say. "But the faster you can get with your multiplication facts, the easier the rest of this unit is going to be.

I'm always excited to see students who use the timed multiplication practice as a tool for filling in gaps in their knowledge. Some students will keep a sheet of a paper and write down the problems they get wrong, and I'll always say what a great idea that is to anyone within earshot. Here's a photo of student's screen - she's using a notepad app to record the facts she gets wrong. Later, she emailed this list of facts to herself for review.

40 minutes

**Overview**

The purpose of today's exploration is to help students generalize some of what they saw on yesterday's Rectangular Gardens assignment, to provide an introduction and point of reference for the idea that quadratic functions can have different numbers of roots, and to make a bridge from the just-completed Systems of Equations unit to these new topics.

Over the course of this activity, students will write a system of equations to represent a problem about a rectangle with known area and perimeter but unknown dimensions. They will graph that system to see that it's not simply a linear system, which will lead to a graphical justification for why some of the dimensions of rectangles on yesterday's assignment were easy to find while others were not. Finally they will see how to use substitution to write our first quadratic function, and they'll see that the roots of that function match the intersection points on the original system.

**Resources and Notes**

How I enact this lesson depends entirely on how independent I can get my students to be. There is plenty of new material here, and plenty that's not too difficult, but that kids will be seeing for the first time. That's why I call today's handout Guided Notes for Desmos Lab - I'll show students as much as I have to, while encouraging independent work whenever possible.

Here is a link to the Prezi that I use for today's lesson notes. I give a video run-through of the Prezi in the previous section of this lesson. Here is a screencast summary of what I'm asking students to do on Desmos.

As we move through the work together, I'll write my own notes on the front board. On the first problem, for example, kids can either write the system or they can't - in other words, this is not a place to ask kids to persevere or try harder - so I just give everyone a moment or two to think about what this task means, but then I provide the notes: a reminder of how to define variables, how to use the Perimeter and Area formulas to create equations, and how to simplify the Perimeter equation by dividing through by 2.

Next, students have no trouble entering these two equations into Desmos, but they may need help interpreting what they see or setting the window to make an appropriate view. Again, I provide guided notes. We discuss the idea that the negative parts of each graph are extraneous, because we're dealing with rectangles with positive area, perimeter, length and width. Fortunately, both intersection points bear that out, both consisting of two positive numbers. We also notice that both intersection points consist of the same pair of numbers: (4,6) and (6,4). We get to discuss how this supports the idea that variable definitions can be arbitrary. X is only the length because we said so, and changing that definition wouldn't change the solution to the problem. I draw two rectangles on the board to further illustrate that point, one standing tall, and one short and wider, but both with the same dimensions, area, and perimeter.

I describe the next step in the Desmos Walkthrough: we change area into a sliding parameter and see what that does to the graph. It's so exciting to watch as kids develop an understanding about what's going on here. Using this dynamic graph, kids can fill out the table on the bottom of the handout. Take a look at this example of student work. As you can see, the graph could use some tweaking, and we're going to need to better elucidate the ideas that the student has written in the table, but she's moving toward understanding the different types of possible solutions to the problem. I've included more student work in a reflection for the next section.

For the rest of this assignment, I try to build on what kids have seen so far and more toward quadratic functions. To do this, we use substitution to solve the system of equations, and then use Desmos to graph the resulting quadratic function. Then, we look for where "Area" shows up in the new function, and after once again turning that into a sliding parameter, investigate what happens to the parabola as "Area" changes. I show the students that we want to watch the x-intercepts. The results match what we saw when we'd graphed the system: that sometimes the roots are unique integers, sometimes they're "messy decimals", sometimes there's only one root (they're equal), and sometimes there are none.

5 minutes

With about five minutes left in the class, I take stock of how far everyone has gotten. As I mentioned above, the ideal is for this lesson to take place during a longer block period, but a regular-length can work as well, as long as I adapt my expectations for how much of the assignment students will finish.

When I see how much progress everyone has made, I tell students to pay particular attention to one or two problems. Maybe it's the last two, if we've had enough time to get there, or maybe it's an earlier problem or graph. There's plenty to learn about what kids know and can do by looking at any part of this activity. Please see my "Student Work" reflection for some examples.

Whatever the scenario, it's useful to take a look at what students have done. When the bell rings I collect everyone's work.