As students arrive, I collect the homework from yesterday's lesson. As the unit on systems begins, we're building background knowledge. The students are developing their own understanding of how to solve problems in two variables, and I'm learning how they approach these sorts of problems. These problems are part of the trove of classic systems problems that we'll see in various contexts over the coming weeks. Each time students learn a new strategy for solving a system, we'll be able to revisit the work from these first few days, and see how to apply the new skills.
Included here is a sampling of what students give me on this assignment. I love teaching systems because there are so many accessible problems to use. This is a unit where students will really recognize the efficient power of algebra for solving problems that they "already know how to do." Think about how seeing this variety of work from your students at the start of a unit can help you plan for the next few lessons. I see a lot of great thinking, and I'll be able to reference this work as we move into the formal approaches to solving systems.
Today's opener (on the first slide of today's lesson notes) is the same as yesterday's. This is a structure that I'll use frequently over the course of the systems unit. I'll give a problem as an opener one day, give students a chance to try it, provide some time to discuss it and to look at student work before leaving it alone for 24 hours. This gives students a time to think about the problem on their own time, which a surprising number of kids will do, given the chance. There are always one or two students who will work on the opener instead of paying attention to the rest of the lesson, but my experience is that kids like that are getting what they need from doing so.
On the second day, I greet students and ask what they've got so far. Some will show me that they're done or that they have a specific question about where they got stuck. I urge these students to lead their groups through what they're thinking about.
Now that students have had time to think about the problem on their own, I'll use this problem to teach a mini-lesson, and I won't leave any questions unanswered. Today, I want students to recognize the need for an ordered list. I want students to take their lists of possibilities, and figure out a structure that will ensure that the list is complete.
I give them a few minutes to work, then I elicit some best practices. "Who thinks that they have an organized list?" I'll ask. Then I'll press students to explain what they did. Soon, it will surface that we have to list possible solutions from least to greatest or greatest to least, so I'll ask which is better. Students will have opinions, but then I'll point out that it doesn't really matter which you choose, as long as you choose one option and stick to it (that's a point that will be echoed when we define variables - when you have two unknown quantities, it often doesn't matter which is x and which is y, as long you're consistent).
Whatever we choose, I make sure that students get this key point: there's a minimum number of each coin, and a maximum. For quarters, the only possibilities are to have 0, 1, or 2 of them. "Can you have more than two? Less than zero?" I ask. "Can we have any numbers in between?" By following this reasoning column by column, we can see that nothing is missing before, after, or in-between the numbers of coins in each column. There are also patterns. As the number of dimes decreases by 1, what happens to the number of nickels? Students are thrilled at the common sense (cents?) of the idea that a dime has the same value as two nickels, and that the patterns in a complete solution indicate just that.
Today's opener is repeated from yesterday, the structure of the homework is the same, and this part of the lesson is no different. I give students another copy of the collaborative problem solving handout, and a new problem called Chloe & Zeke Part 1, on the second slide of today's lesson notes.
Just like yesterday, it's a think pair share structure. Students have a minute to record the problem and five minutes to work on their own. Then, I ask them to pass their work to the left, and to provide feedback on the work of a classmate. After that, they share out.
The problem itself is different from yesterday's problem in a few important ways. While yesterday's ducks and cows problem can be represented by two linear equations in standard form, one of the equations for today's lesson will be in slope-intercept form. Eventually, students will recognize that substitution might be the best option to solve this problem, but that elimination might be better for the ducks and cows problem. Unlike yesterday's problem, where it really didn't matter, there is definitely an advantage to making Chloe's age the dependent variable. We'll get to all of this over the next few days, but I wanted to lay out here some of my thinking behind starting the unit with these two problems.
When it comes time to share-out and debrief on this problem, there are lots of opportunities to build toward algebraic ideas. Here is the work of a pair of students. As I describe in the Video Reflection for this section, there are two ways to guess on this problem, which provides an amazing opportunity, and it's what I look for as I see what students are doing on this problem.
It's a soft transition from the collaborative problem solving part of the lesson to this one, and I'm flexible with time, so what happens next differs from class to class. At this point I start saying that we're getting a head start on tomorrow's lesson.
We're going to start analyzing yesterday's "Ducks and Cows" problem and today's "Chloe and Zeke" problem with some algebra. Tomorrow, in the computer lab, we'll use Desmos to graph both problems. The minimum I hope to accomplish today is to introduce the idea of carefully defining variables, and using them to write algebraic rules. I don't force the algebraic analysis on the one hand, but when students produce an opportunity like the one I've outlined in the previous section, I'll run with that.
This photo is actually from tomorrow's lesson, but some classes might get this far today. Today, I'll lay foundations for that. As with the openers that span two class periods, I don't rush to a solution to any problem. I want kids to have space to try things on their own, and tomorrow's graphing activity will provide a dramatic reveal of the solution. For a description of how I use this problem to move into algebraic representations and graphing, please see tomorrow's lesson.
Tonight's Homework is just like last night's. There are two problems that can be solved by guess and check or by algebraic methods. The first problem is just like the ducks and cows problem, except with ants and spiders, and kids get a kick out of that. (They've also pointed out that technically a spider is not a "bug" so this is really a trick question. I'm not sure where I stand on this issue.)
The second problem is demanding in terms of the language, and it has three variables, but a rather simple solution. Problems like this, in three variables, are great to revisit when we study substitution.