Why I give my students a system of 6 equations to solve on Day 1 of school.

One bit of advice I give student teachers when they observe a classroom is to always ask the teacher when they see something that they don’t understand.  If that teacher is good, they will most likely talk for a long time about the dozens of reasons they made that choice.  For example if you wonder why a teacher is letting a student keep their head down during class, ask.  I once asked this question of a peer and the teacher spent the next 10 minutes telling me about this students living arrangements (homeless, 4th school this school year….).  Good teachers have reasons for everything they do. (It is why our job is exhausting – the amount of information we process & the number decisions we make in just one class period is more than non-teachers can imagine making in a week of work). What can be hard in training new teachers is all the invisible decisions we make in our classrooms and helping them see those.

When I share lessons with other teachers I know they will never come close to understanding the invisible choices I made as they look at my slides, assignment sheets – even if I include a written lesson guide.  I know it is possible they will judge a task I selected or how I selected to start it – not knowing I have a ton of thought put into everything I put in front of students.  When I look at lessons and resources on other teachers blogs I know that even in a blog post they can not give me the hidden magic of the entirety of their intentions with what they present.  In the last 3 years I’ve co-planned lessons daily with a couple of different people.  This commitment to talking out loud about what we will do and why we will do it that way is what impacted my student’s success with what happens in my classroom more than almost anything else I do.  Between us we make our invisible intentions visible we are able to make changes that are best for our students.

I say all this because on my blog I know there are ton of things here you will never understand unless you asked me in person about them and let me start talking – and I can  talk a lot.  SO, I thought I would give you one example from a question I got from a math teacher peer in my district this week.  He asked me, Why do you use this slide with students on day 1 of Advanced Algebra?


What the teacher really wanted to know is ‘why in the hell are you asking your students to solve for six unknowns on the very first day of school?‘.  I was so glad he asked me.  Here is what I said to him (remember, I can say a lot!)…

  • My goals for teaching students the first week of school have little to do with our state math standards. (gasp!).
  • I first used this problem 20 years ago while teaching out of the Interactive Mathematics Program (IMP) text.  Part of IMP is to use a problem at the start of each unit that students play around with but do not have the skills yet to solve.  On day 1, Capturewe would hang a poster on the wall – we called it the UNIT PROBLEM – and point at it every day and ask students what math we were learning that day that would help us solve the problem.  The poster of the unit problem was way more engaging for students than the learning targets many leaders ask teachers to post in their rooms today.  When I went to middle school to teach and used a different text I found that one thing I missed from my IMP days (there were many) was having a central problem in each unit I could refer to each day of the unit.  On day 1  of school, one message I want to send my students is ‘this is what you will be learning to do this year’.  On day 1 I don’t want to post wordy learning targets (ick) – I want send the message our learning will be centered around problems and tasks.
  • I use this problem because I want to do something mathy on day 1 of my class.  I want to convince my top performing students (or those that think they are top performing) that we will be doing math that they don’t know how to do yet.  Why do I do this?  I do this to prevent some of the – at times annoying – things top students tend to do or think.  I am sending a message that you can’t solve everything with guess & check.  I am sending a message that even if you think you are smarter than others in this class, there will be a lot of math this year that you don’t know how to do yet.  I am sending a message to students who’ve been bored in other math classes that they will see things in this class this year that will challenge them.  I am getting the students already addicted to the feeling of that comes from solving math problems excited that they will feel this in my class.  I want them to go home and tell their parents that this class will be challenging for them. On day 1  of school, one message I want to send my students is ‘we will do challenging math in this class’.
  • I use this problem because even though I want students to feel like they are doing math on day 1 I don’t want to shut down my students who are afraid of math on the first day of class.  I do not ask my students to solve this problem on day 1.  In fact I don’t have them even try to solve it.  My goal is for students to not jump into their fear of inadequacies of math day 1.  After doing the things I will describe below with this problem I confidently announce to the class “Everyone of us in this room will be able to easily solve this problem later this school year.  I don’t care if you don’t believe you are good at math.  I promise you, you will tell me that this problem is easy before the end of the school year.  I am excited you are in this class.  We will all learn a lot of math this year. More importantly, this year all of you will feel successful in math.“.  On day 1  of school, one message I want to send my students (particularly those who are feel they can not do math) is ‘don’t worry – you will be able to be successful in this class’.
  • On day 1 I want to start using routines I will use all year.  One of which is my favorite three questions to ask in the mathematics room – questions I will ask them almost every day of class during the year –> What do you notice? What else? What do you wonder?.  Many of you reading this know these questions from The Math Forum and from Annie Fetter’s Ignite video.  The way I get away with using a problem on day 1 that seems impossible to solve without loosing any of my students who fear math is to present the system of equations and ask only “What do you notice?”.  I hand out a copy of the system for students to glue into their notebook.   (You can find it HERE:  system of 6 equations for notebook).  I then ask them to write down at least 10 things they notice.  I then ask the class to shout out what they notice as I record their noticings.  I then ask ‘What Else?’ a ton of times until I have a list of 10-30 things recorded.    On day 1  of school I want to start right away using the routines they will see all year in this class.  
  • I use this question for formative assessment reasons.  I want to know what my students enter my classroom knowing.  Here is a list of things my students have said after using the noticing/wondering routine in class with this task:
    • I see numbers.
    • There are letters
    • The letters are variables.
    • There are 6 equations.
    • There are 6 variables.
    • There are odd and even numbers.
    • There are plus and minus signs
    • Some of the coefficients are negative.
    • This problem is an algebra problem.
    • I bet we will have to solve this.
  • Students start naturally talking about what it means to solve a task like this.  They shout out words like substitute, eliminate, graph…..They wonder if there is a solution to the problem.  On day 1 students have already said and/or heard math vocabulary they’ve used in prior math classes. I learn a lot what students know and don’t know from prior math classes on the first day of class using this task.   On day 1  of school I want my students to open up the part of their brain that was dominant for much of the summer and begin using math vocabulary that will be integral to our work together this school year.
  • The reason I have my students glue this into their notebook on day 1 is I can then refer to it throughout the school year and ask students if we know enough math to solve this yet.  About 1/3 or 2/3 of the way through the school year we get to a unit about solving systems of equations and students learn about matrices.  I then tell them we are going to solve the problem from the first day of class.  I have them turn to this problem in their notebook (by the way – I call this the ‘blue’ problem – I ran it off on blue paper and students can find it faster when I called it the ‘blue’ problem) and have them solve it with matrices.  I love using this problem on day 1 because later in the year students have such closure in their brains when we do solve it.  

I love this problem.  I’ve never yet had a single student solve it before we learn about matrices.  This task has challenged students without scaring away my most math phobic students.  I have loved that on day one of class some students are wondering what a solution would look like (point, line, plane, something else) if they did solve it.  I love that I kind of scare my students a bit with this problem knowing that all of them will feel successful later in the year when they solve it.

In closing – I challenge us all to ask peers about things you notice about their classroom.  There is a lot of hidden brilliance that needs to be uncovered so we can all learn and grow. I challenge you to look at each item you put in front of students.  Here is a quick litmus test for you about the potential quality of the task for your students – If someone were to ask you about the item and you can’t talk a lot about why you selected this item – then it probably should not be in your lesson.  Good teaching has intention.  Student success on the tasks we give requires us to plan.  Don’t just take a problem from the book your school assigned you without asking why you are using it.  Make sure your reasons for using what you use are both mathematical and also address the social needs of your students.


Sara VanDerWerf

I am Sara Van Der Werf, a 24-year mathematics teacher in Minneapolis Public Schools. I have taught math in grades 7-12 as well as spent several years leading mathematics at the district office. I currently teach Advanced Algebra at South High School and I'm also the current President of the Minnesota Council of Teachers of Mathematics (MCTM). I am passionate about encouraging and connecting with mathematics teachers. I'd love to connect via twitter.  Join the community.  Tweet me @saravdwerf.

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