Today's opener (on the second slide of today's lesson notes) is a sum and difference number riddle, just like the first six problems on last night's homework assignment. As on problems 4 through 6 from that assignment, the solutions are not whole numbers. By starting class with this problem, I give students one of two opportunities: either to show off what they've figured out about solving problems like this, or to say, "Yeah, wait, what am I supposed to do on a problem like this one?" It usually happens pretty naturally that kids get to talking about this problem and the second group of kids get help from the first.
As soon as a student volunteers to show everyone what they know on the board, I let them go for it. If that doesn't happen, I give students a few minutes to try this problem and then to take out their homework before I ask someone to come up and show us what they've got.
When it comes to sharing out, it doesn't matter whether we use the opener or a problem from the homework. The opener leads into a debrief of the homework, and often kids will just make the transition on their own. So when a student shares a solution strategy like this for one of the homework problems, we can then see if it works for the other problems. What I love about this student's work is that he's using elimination to solve the system of equations. It will still be another week until I officially teach that topic, but usually I'll take a moment here to show everyone what the algebra looks like, matching the two step by step.
We compare the first three problems on last night's homework (they're easier than the opener, because they have whole-number solutions) to the next few (with their decimal solutions) and note that as soon as you have an algorithm (in more kid-friendly terms a "process" or a "strategy") it's going to work, no matter how "easy" or "difficult" the numbers might be.
Once students come to that idea, then they've got a justification for algebra. That's a great theme of today's and tomorrow's lessons: at this point, kids really start to ask for the algebra. It's so cool when it comes from them.
In terms of collecting homework, I do collect these assignments, but I don't put a hard deadline on anything. I tell students to finish #5.4 and #5.5 as soon as they can. My main motivation for collecting the work at all is that I take it out of circulation: I can ensure that students who are still grappling can't just grab the completed work of someone else, and that this maintains urgency to get it done.
Today's activity might take a little more than the time I'm allotting it here. Depending on how long the opener and homework debrief take, and depending on how comfortable my students are with everything we've done already, what I describe during this section of the lesson might extend into tomorrow's class. For continuity, I'm putting it all here, but with most of my classes, it's likely that I'll need a little more than 25 minutes to get it done.
About the Activity
Today we're going to revisit the "Perimeter of a Rectangle" opener from two days ago. As I have a few times in the last two weeks, I introduced this problem at the start of Tuesday's class, but did not rush to a solution. Now we're going to solve it in three different ways. I call this "Guided Problem Solving" because the lesson moves back and forth frequently and seamlessly between instruction and work time.
Students get this Three Ways to Solve the Rectangle Problem handout, which provides space to solve the problem by Guess and Check (that was Student Learning Target 5.1), by Graphing (SLT 5.2), and Substitution (SLT 5.3).
I start with a review of learning targets. The first two are on the back wall of my classroom and SLT 5.3 is new on a slide that I project to the class. I say that today we're going to learn about an important algebraic idea called substitution, and we'll see why it's such a useful tool.
To guide the lesson, I put today's lesson notes on Prezi. What I like most Prezi is that it's so easy to upload a .pdf document and then to "fly around" it, zooming in on different parts as needed. You can see what I mean here.
I give students a moment to find any work they've already done on this problem, and then we move through it, one strategy at a time.
Solution Strategy #1: Solve by Guess and Check
As I mentioned in the opener, the purpose of today's lesson is to provide a justification for why algebraic methods exist. To that end, I provide two options for guessing and checking to solve this problem. In the first table, we can come up with dimensions that give us a rectangle whose perimeter is 448, and then check whether or not the length is 2.5 times the width. In the second, we can try pairs of numbers such that "the length is 2.5 times the width" before checking whether or not we get P = 448. Each table begins with two examples that should clarify what a "good guess" looks like, and I model how to check these before letting students try a few more. I think the second way is more efficient, but I want students to have space to come to that conclusion on their own, and starting with a less-efficient method is a way to make that clear.
Two things should happen here. First, students should see that, although guess and check can work, it's not always an efficient method. Second, students have a chance to build their algebraic reasoning about this problem by varying their guesses, which will then help us to define two variables and write equations to represent the two constraints given in the problem. In each table, the two guess columns end up being a list of points to satisfy one constraint, and the check column becomes place to see an equation representing the other constraint develop.
I tell students that it doesn't matter whether or not they get to a solution within the limited space of each table. If they can, that's great! I tell students to fill in each row of each table, and then to assess how close each successive attempt comes to being right. Whenever someone comments on one table being more useful or user-friendly then the other, I ask them to expand on that, and the conversation about efficiency can take off from there.
Solution Strategy #2: Solve by Graphing
In order to solve this problem by graphing, we'll need two equations that we can graph. That work begins beneath the two guess-and-check tables, as we define the two variables. I use the second guessing table to show that's a little easier to let x represent the width of the rectangle, because then we can just multiply by 2.5 to get the length, y, which naturally yields the equation:
y = 2.5x
With the variables defined like that, we can just use the formula for perimeter to get the other equation:
2x + 2y = 448
There are a few moves that might make each of these equations a little easier to graph. I show everyone that the first equation can also be written as:
y = (5/2)x
and that writing the slope as a fraction will make it easier to plot points on our graph. In the second equation, we can divide through by 2, and get:
x + y = 224
As some students have already noticed, asking for length and width such that the perimeter of a rectangle is 448 is basically the same thing as asking for two numbers with a sum of 224. The algebra confirms that they're right. (That's an idea that will come up once again in my Quadratics unit, when we use area and perimeter problems to move into factoring.)
Some students will also point out that we can also write the perimeter constraint in slope-intercept form:
y = -x + 224
and that's what I'll reference when we graph the line.
Throughout all of this, it's important to note that we've already made two sets of points in our two guess-and-check tables, so we can always use those guesses as coordinate pairs. Usually, students are comfortable enough with graphing lines in slope-intercept form that we won't need those "points", but it's nice to know that they're there and can help us check the accuracy of our lines later on.
Turning to the back of the problem solving handout, there's a 30x30 first quadrant on which to graph these two lines. Our next decision is about how to scale the axes. I use the perimeter equation to help us make that decision. The intercepts of this equation are (0,224) and (224,0), so we'll want to be able to see both of those. Noting that, students are quick to conclude that we should count by 10's on each axis. Everyone takes a few minutes to complete that task. I do the same. When I work on the board, I'm modeling craftsmanship and attention to detail for my students (see this image: Plotted Points). I want them to feel like this is easy. Rather than writing every multiple of 10, I skip-count and only write the 50's. I ask the class, "What am I counting by here?" I want them to note that even though I'm only writing 50's, I'm still letting each line on the graph count for 10.
When that's done, we can plot some points. I show students that when both axes are equally scaled, we can use the slope and the method of counting "up and over" to plot successive points. When we "rise 5 and run 2", we're really going up 50 and over 20, but 50/20 reduces to 5/2, so everything works. (Tomorrow, we'll have an opportunity to investigate what happens when the axes are not equally scaled.)
With the points on the graph, we finally see that we're going to have to connect the dots to see the intersection. Once we do, the resulting intersection point should make plenty of sense!
Solution Strategy #3: Solve by Substitution
So finally we get to the algebraic solution of substitution, and it's always such a beautiful moment. I say that I want everyone to see something great, and I write this on the board. After all the work we've done, I want this to look simple and elegant to the kids, and it always does just that.
I note the difference between y = 2.5x and y = (5/2)x. As we've seen, the latter is better for graphing slope. The former, however, makes the calculations nice and easy however, and this gives students a chance to appreciate why we have different ways to write the same number.
Upon seeing this, my students have been known to beg to learn about substitution! "Why didn't you show us this before?" they say. I reply, "Now you know why algebra was invented!"
And now that they know, we can joyfully move toward mastering all the moves that make algebra such a powerful tool.
I note at the beginning of the previous section that the "Three Ways to Solve a Problem" activity will often take us beyond the end of today's class. Just to be prepared, however, I have a few extra problems ready to go. They're at the end of today's lesson notes.
The "Engel" problem will be used as an example during tomorrow's lesson.
The "Phelisha" problem will then be used as part of a check-in quiz on SLT 5.2.
You should adapt the pacing of this pair of lessons to your own classroom. Please feel free to let me know how it goes!