Three Days of Review, Problem Solving, and Achieving Mastery

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SWBAT assess their strengths in solving systems and work on improving where they need it.

Big Idea

Time is structured to encourage student autonomy and decision-making, and it's outstanding to watch students rise to the challenge.

About this "Three Day Lesson"


5 minutes

Monday's opener is on the first slide of the lesson notes, it takes us back to the Ducks and Cows problem that opened the unit.  I give kids a few moments to work, and then we discuss the problem.  It's great to give kids a chance to revisit this problem with the all the algebraic tools they've seen over the last few weeks.  I ask the class which method will be best for solving this problem now, and although a few students will want to use substitution (and that's great if they do!) we usually decide to run through an elimination solution to this problem.  Whatever we do, the work stays up on the board for reference.

For Tuesday's and Wednesday's openers, nothing is prescribed and there's more of a soft start.  I might put a problem or two on the board that I noticed a lot of people struggling with the previous day, but I'm most likely just to give kids space to get going on whatever they can.

Mastery Tracker and Getting Started

10 minutes

To frame this week's work, I distribute mastery-based progress (here is an example) that include all of the Student Learning Targets for the year to date, and this "Mastery Tracker", both of which I describe in this video.  

I use slides #4 through #7 of the lesson notes to lay out the plan for students.  I show them how to record their current Unit 5 grades on the Mastery Tracker, and I describe how they can plan for the week.  "You should focus on your lowest grades first," I say.  "If you have a 1 or a 0 on any of these four learning targets, then that's where you should start."  I add that no one has grades yet for SLT 5.4, which is about solving systems by elimination, so that's a likely starting place for everyone, but there are plenty of kids who need some work on at least one of first three SLTs.  Slide #7 highlights the three places that I expect everyone to start.  "I want everyone to have at least a 2 on every learning target," I say.  "So to begin, you should pick one of these three assignments."

For reference throughout the week, all four Unit 5 learning targets are on the back wall of my classroom.  Everyone is going to have three opportunities to demonstrate what they know: they can work on whatever they'd like for the first three days of the week, they'll have a chance to solve some open-ended problems on Thursday, and they'll take a multiple-choice exam on Friday.  At this point in the year, students know that for each of these content learning targets, it's the highest grade that counts.  This means that no one needs to be anxious about any one assessment, but that everyone needs to get something done to justify their grades.


SLT 5.1: Guess and Check

Most students have already achieved a 3 or a 4 on this learning target, because we spent ample time using guess and check to frame our work at the start of the unit.  For anyone lacking evidence, I give students another chance to work through the Number Riddles Problem Set.  For students who have struggled, taking a new, focused look at number riddles like these can really make the rest of what they've seen click.  I don't expect to see a lot of "pure" guessing and checking on these problems.  By now, student work will be more of a hybrid of formal and informal algebraic strategies.  When I see what a student can do here, I can help them plan their next moves.

The "Mastery Problems" for this learning target that are referenced on the mastery tracker come from the Guess and Check chapter in the fantastic Crossing the River With Dogs textbook.  I have a few copies of this book floating around, and whenever a students asks for the Mastery Problems, I direct them to a few problems there.  

As a side note, I heartily recommend getting your hands on a copy of that book if you can.  It's a rich trove of problems, and I've used it successfully with upperclassmen who have struggled previously in high school math.  The book is also full of examples of student dialog about problem solving, and always inspires invigorating conversations among teachers. 

SLT 5.2: Graphing Linear Systems

For any students to not yet achieve a 2 on this learning target, I've used Infinite Algebra to prepare a set of eight systems to be graphed.  We've spent a lot of time on this learning target as well, and at this point, most students are confident enough graphing lines.

Where many students need work is on solving problems by graphing, and that's what this Solving Systems by Graphing handout is for.  I tell students that I know they can probably solve problems #1-4 by methods other than graphing, but that I want to see what a perfect graph of each looks like for each problem.  This proves to be a useful review exercise for many students, as they continue to understand that intersection of two lines gives us the solution to a problem with two variables.

The fifth problem has several questions, and using a graph to answer each one helps students to review some of the ideas about what makes a graph especially useful.

This year, most students focused primarily on substitution or elimination as the week progressed, so it was never necessary for me to set up the "Mastery Problem Set" for this learning target.  Graphing plays a big role on Friday's multiple choice exam, so that's a great opportunity for a lot of kids to raise their SLT 5.2 grade from 3 to 4.  

SLTs 5.3 and 5.4: Substitution, Elimination, and Problem Solving

Using substitution and elimination to solve problems will comprise most of this week's work for most of my students.  I use the same progression of work for each learning target, 5.3 and 5.4.

To earn a 2 on each learning target, students complete 10 problems generated by Infinite Algebra.  To limit the amount of paper used, I print two-sided handouts with "Substitution Practice" on one side, and "Elimination Practice" on the other.  I don't make too many copies of this, because it makes sense for a lot students to jump straight to the "Advanced Practice" for each.

I make two-sided copies of the Advanced Practice for SLT's 5.3 and 5.4, with SLT 5.3 Advanced Practice (Substitution) on the front, and SLT 5.4 Advanced Practice (Elimination) on the back.  Both of these start with some basic examples.  For example, on the Substitution side, I challenge students to solve #3 in their heads, and it's a challenge that students are excited to see that they can accomplish handily.  The real purpose of that prompt is that it helps me see whether or not students understand the idea of substitution in the first place.  On both of these practice sets, the problems get successively more difficult, and provide examples for great conversations as the week continues.

For problems #15-26 on the "substitution" side, I only ask for students to justify their choice of how they'd go about solving each system, which is another way to get review and synthesis-oriented conversations going.

On the back wall of the classroom, I post example solutions to four problems on each side of the page (#7, 9, 11, and 13 from the substitution side, and #14, 18, 24, and 25 from the elimination side).  I am offering increased grades for anyone who finishes this work, but that's really just an extrinsic sort of motivator to get kids going.  What I really want is to help them review these ideas, and to feel confident that they're getting it, even when the numbers make things a little more difficult.

I try to push all students to get to this Mastery Problem Set by the end of Tuesday's class.  If kids can solve these problems, then they've really got a grasp of what I wanted them to learn during this unit: setting up a system by defining variables and writing equations, then choosing the best method for solving that system.  

Work Time

100 minutes

Here's what typically happens: by Tuesday, each of my classes are split into two parts: students who have kept up and are working hard to get through these problems before reviewing for the exam, and students who are filling in gaps in their knowledge by working on the practice exercises for each learning target.  It's exciting to see 20, 30, or 40 minutes pass with everyone working hard to get what they need.

I set a daily agenda with reminders, but as I describe in this video, I really try to situate myself at the back of the room, setting up shop at one table, and telling kids to come to me with questions, requests for work, and to show me what they've done.  I want to encourage autonomy in my students, and I do so by making them the ones who move around and decide what to do.

Each day, the front board will indicate variety of examples that we've worked on together that day (I can't sit still for that long, after all).  Students will work in their groups, but as you can see in the "working" images here, often students sitting side by side will be engaged in different work.  Some students will be at the board working out problems while others are at their seats.  Other students will check their work against what's posted on the back wall with the learning targets.

I feel most successful when students are able to talk about what they've learned, when they say things like, "I've got substitution down, but I need some help on elimination."  When we have these conversations, I remind students that they can prepare cheat sheets to bring to the exam.  "Anything that you feel confident about is great," I say.  "Anything that you're feeling like you might forget it, or any problems that felt especially difficult, that's what you should work on tonight, and that's what should go on your cheat sheet."