The Most Important Algebraic Idea of the Year

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SWBAT use substitution to solve systems of linear equations, and they will understand what a powerful tool algebraic substitution can be!

Big Idea

At the center of algebra is the idea that if this is the same thing as that, then we can replace that with this, and use the results to solve a problem.

Opener (and opener, and opener...)

15 minutes

New Seats

Before the late bell, I post today's new seating arrangement on the board.  It's based on the results of yesterday's mastery quiz; students are seated with classmates at similar levels of mastery on SLT 5.3: 

I can use algebraic substitution to solve a system of equations.

When we get to work time, students will work together to "level-up" from wherever they are to the next.

Three Openers

On today's agenda, I write, "Opener (and Opener, and Opener)" so anyone paying attention will know that today's opener is going to happen three times.  On the first three slides of today's lesson notes are number riddles that can be efficiently solved by substitution.  The first two have exactly the same structure, just with different numbers.

I give students a few minutes to try the first problem, and then I have them guide me through the steps as I record my work on the board.  We define variables, and write equations, starting from the word "is" as our equals sign.  Then we solve the system.  The solution to the first problem looks like this.  When I write these notes, I'm really trying to model what work can look like on the page as much as I'm modeling the mathematics.  I tell students to pay attention to how their equal signs line up, and even to the spacing between the lines.  "You don't want to scrunch your lines to closely together," I say.  "Make sure your work has room to breathe, because that will make it easier to read later."

When I introduce the second opening problem, I ask students to compare it to the first.  Everyone notices pretty quickly that this is really the same problem as the first one.  To further emphasize this point, I erase the values from the first problem, to form a sort of template, before filling in the blanks with new values.  Some students are a few steps ahead of me, and have already completed the problem on their own.  I want to make sure it's clear to everyone else that the only thing that might make this problem feel difficult is the nature of the numbers involved, but that the algebra for each of these first two problems is exactly the same.  After the initial shock students feel about seeing the numbers in "Version 2," many of them are grateful that the tool can feel so simple to use.

The third problem is a bit more demanding, but once again, I adapt the structure that's already on the board to this problem.  If kids can do this problem on their own, then we're in great shape.  If they need some help writing the equations, then that's what I'll help them do.  When we're done, we have an example of a system that is best solved using substitution.  I tell everyone that this is the most important idea I want them to understand about algebra.  We can take a demanding word problem in two variables and use substitution to come up with an equation in one variable that's not too hard to solve.  

As we transition to work time, I leave this equation on the board and allow students to finish up the problem.  I say that I'll check the work of anyone who can finish this up, or that students have the option to get to work on whatever they'd like.

Leveled Practice: Substitution

15 minutes

For the second consecutive day, I give students the middle part of today's class to practice solving systems by substitution.  Today, students are grouped by the level of mastery they demonstrated on yesterday's quiz.  As return the quizzes, I make a pretty strong suggestion to each group about what they should work on now.

Around the room, I post some exemplary work from yesterday's quiz, as well as an example or two from each of the leveled worksheets that students are working on now.  I tell students to get up and take a look at examples if they get stuck.  I find that encouraging movement in small ways like this can really help to build a productive classroom vibe.

All of today's practice worksheets are made on Kuta's Infinite Algebra software.  Textbook problems or other practice sets will work fine here, but I like the way I can customize the work on Kuta.  For example, the "Level 4" worksheet starts with two special cases: a system with infinite solutions and a system with no solutions.  It ends with problems in which there are no coefficients of 1, which will take us into elimination next week.

As students work, I really try to get them talking to each other.  When I see that a student is developing new expertise, I encourage them to share it with their table-mates.  If a student has a question about a problem, I might refer them to classmate who can answer it.  If two students are up taking a look at the same example problem, I'll ask them to discuss what they notice, what they already knew, and what's new on this problem.

In every conversation I have today, I continue to reiterate the idea that substitution is such an important tool, because it really gets at the heart of what makes algebra work.

Two Mastery Problems

13 minutes

Now that students have had time during the last two lessons to practice with their algebra skills, it's important to remember that mastery is being able to solve problems, not just run the algebra.  I tell students that they have the opportunity to demonstrate that kind of mastery now.

There are mastery problems on the sixth and seventh slides of today's lesson notes.  I tell students to solve each of these problems on loose-leaf paper, and that I have graph paper available for anyone who wants it.

The first problem is a number riddle like the the three openers.  It's important that it's not exactly the same problem, however: the word "sum" replaces the word "difference," for example.  It's interesting that there are always a few kids who can do perfect work except for noticing that change of words, and that leads to some neat conversations about attention to detail.  I leave the first problem on the board for a few minutes, until I'm sure that everyone has seen what they need to see.  Then I switch the second problem.

The second is more open ended, and allows kids to apply their knowledge that different representations will have their advantages.  I don't say anything about solution methods for either problem.  While the first problem is pretty clearly suited for solving by substitution, the second is not.  It's great when kids ask, "Wait - I want to graph this!  Can I?"  I tell them that of course they can!  Ideally, students will see that a graph can help us answer both of the questions posed by this problem.

As always, an assessment like this is a tool that helps me see what students know, and it helps me collaborate with students to define our next steps toward mastery.  The student work I've included here isn't all perfect.  When I look at this work, it becomes clear what students know and what they'll have to learn next.  Please see tomorrow's lesson to see how I look at this work with my students.

Tomorrow, we'll debrief on this work as we continue to move back and forth between algebraic practice and problem solving.  Tomorrow's class will start with another mastery quiz for just the algebraic part of solving systems by substitution.