As students arrive to today's class in the computer lab, I return the graphing quizzes they took yesterday. I use Desmos to post the quiz solutions on the board, so everyone can check their work. As you can see in the solution image, this quiz makes it very difficult to find the precise coordinates of each solution point. That is by design. I want students to get the idea that it's useful to make approximations for solutions like these, and to understand that they can expect a system of equations to yield a "messy" solution like these. With that in mind, I do make the numbers a little bit nicer for students on the next such quiz.
I explain how the quiz was graded. Four perfect graphs earned a "4" on SLT 1.7, while four perfect graphs and appropriately labeled intersection points earned a "3" on SLT 5.2. To earn higher than on SLT 5.2, students will have to show that they can use graphing to solve and interpret the solution to a problem in context. They will have a chance to do that soon.
This quiz frames today's work and helps to inform students where they should begin. Using Delta Math, students will practice graphing lines, changing equations from standard form to slope-intercept form, and then graphing systems. Kids actually love this - after days of problem solving, they like having the chance just to practice some algebra. I rely on this ebb and flow of a class to keep kids engaged.
With yesterday's quiz as evidence of what they need to do today, some students will start by completing a challenge that will earn them a "4" on:
The settings on Delta Math make it easy to define what mastery might be for a given skill. I can set a bar for students to get any number of problems correct, and I can decide if there should be a penalty for getting one wrong. On a foundational skill like this, I'll often require five consecutive problems correct, with a penalty of going "back to zero" if they get one wrong. The software keeps track of each student's personal best: so if they get three right before getting one wrong, there's a record of that.
Today, I take it step further by asking for ten consecutive correct answers. Once it's done, this is a huge morale booster for kids, because it sounds hard at first. But it's just scaffolded enough that everyone is able to do it, and I've often seen kids parlay their success on a challenge like into even better things.
The idea that completion of this challenge can count for mastery is a little bit misleading, because earlier in the year I was a more rigorous in my definition of mastery of this target. I required students to be able to graph lines that were not given in slope-intercept form, and I require them to label and scale their axes in the context of a problem. But, even with that in mind, I choose to be a little more lenient here. I want kids to know and feel like they're getting better at something. And if they don't have slope intercept form down at this point, it's time for that to happen. As we move forward through this systems unit, there will time for them to solve problems in context and to graph lines that are originally given in standard form. Without this morale boost and the foundational knowledge of graphing lines in slope-intercept form, the rest will remain quite challenging.
In practice, this challenge makes for some high-spirited fun. There's just enough of a game to this that kids are super engaged. And using a grade as a carrot, if used sparingly (as with any teaching tool) can yield great results.
Once they get it out of the way, that good vibe propels kids into what comes next. There are always one or two kids who take just about the entire period to finish this task. Often, these are students who feel success too infrequently in my class. I've seen some buzzer beaters - kids barely beating the bell - but with that can come a renewed commitment to the class. That energy is what I'm looking for!
As kids get 10 in a row, I publicly credit them by publicly updating my gradebook on the projector at the front of the room.
The next skill that students will practice is converting linear equations to slope-intercept form from standard form. As students complete the graphing challenge, I invite to the front of the room for a standing mini lesson. We stand at the front board together in small groups, and I show them how to "solve for y" in these equations. Now that they've proven to themselves that they can graph lines in slope-intercept, students buy in when I tell them how important it will be to be able to change the form of such an equation.
The algebra is not too hard here, kids notice, but we have to pay attention to details: misplacing positive and negative signs and making mistakes while reducing fractions account for 90% of the errors students will make on this exercise. Unlike the first exercise, I require that students solve five in a row here.
What's beautiful about the systems unit, and its placement rather late in the year, is that it requires students to review and apply a lot of what they've seen this year. Both of these skills have "been taught" earlier in the year, but not all students have learned them. It's so important to be honest with myself about this. Do I wish that every student was perfect at graphing lines and transforming equations? OF COURSE! But I can't ignore it if they can't, and if they're finally really ready to get this now, which a lot more of them are, then that's what we get done.
After practicing graphing and changing the form of linear equations, students apply both of those skills to solving systems by graphing. Here's a video description of what students on the rest of today's assignment.
At the end of today's assignment are six modules about substitution and elimination to keep everyone who's already mastered the earlier stuff going. Because these problems are multiple choice, there's another option for solving these - Guess & Check, of course! - but I don't mention that to kids.
With a few minutes remaining in class, I distribute tonight's Homework 5.5, which consists of a series of number riddles. In the coming days, these problems will help to distinguish between informal solution methods (like Guess & Check) and more formal algebraic methods. I don't say too much about the assignment when I give out to kids, but we will spend some time looking at one or more these problems tomorrow.