When Knowing 3 * 2 = 6 Isn’t Enough

We recently started our expressions unit where I introduced kids to the joy of Algebra Tiles and modeling expressions.  I was feeling pretty good about how things were going until it was time to begin the distributive property.  Out of pure curiosity at the beginning of class one day I asked the kids to grab some integer chips or Algebra Tiles and model two times three.  I would be lying if I said I wasn’t shocked about what they did.


This was by far the most common model.  The discussion went a little like this:

Me: How many chips are in front of you?

SS: Five

Me: But what is two times three?

SS: Six

Me:  Then why do you only have five chips in front of you?

SS: Because you said to model two times three.

Confused, we had a quick discussion about the meaning of multiplication being about making groups.  We talked about how two times three really meant either two groups of three or three groups of two.  They seemed to get it so I moved on like any other terrible teacher and later tweeted the picture.  Someone on Twitter challenged me to ask them another modeling question the next day but this time with a fact they didn’t know.  I said challenge accepted, I mean after all I had just told them what multiplication meant they surely would get it right tomorrow.

The next day I asked them to model fourteen times eight.  On a positive note there were kids who completely nailed the model this time and I would have thought I was really good teacher…

20140925_120113 20140925_120139 20140925_132540

Except the other half of the models looked like these:

20140925_132604 20140925_132513 20140925_120123 20140925_132521

Now interestingly enough, even the kids who made these models incorrectly got the correct answer because off to the side of their table they had worked the problem out using the standard algorithm but still really not knowing that 14 times 8 actually means 14 groups of 8 or 8 groups of 14.  We did some more investigations into this using both multiplication and later division until it seemed like we had a good grasp of what was going on.  The kids conversations during this time were priceless:

S1:  Did you know that is what multiplication meant?

S2: No but do you think it is weird she is teaching this to us?

S1: Kind of but I think she might be teaching it the right way.

Later I tweeted a lot more of the modeling frustrations to mixed reviews.  Of course math teachers were mainly supportive but others questioned why a kid needed to know why 2 * 3 = 6 as long as they memorized the fact.  I guess for me that just isn’t enough.  I think kids need conceptual knowledge of what they are doing in math class rather than just blind trust that the facts they memorized are correct.

These are the kids that when asked to model 2x end up modeling x +2 as they do not understand the difference in the two.  I see kids hit the wall with this a lot with the distributive property.  If they do not understand multiplication as an area model they almost always struggle with understanding how to distribute which I am sure if how we ended up with teachers asking kids to memorize FOIL rather than really asking kids to visualize what is happening.

20140925_133255 20140925_133403

Trust me when I say that I have a lot left to learn as a math teacher, probably more than is possible before I retire.  But I do know and believe that pushing kids to understanding mathematics conceptually rather than blindly memorizing facts and rules is where my heart lies.  I don’t think I would or could be a teacher if 2 * 3 = 6 was good enough.

My 1st Grader Changed My Thoughts on Homework

My son is in 1st grade this year and it has been an eye opener for me.  When I first started teaching I was one of the really good math teachers who assigned 30 homework problems a night on a bad night or on a good night #2-30 even only.  I cringe now when I think of those early homework assignments.  They were so poorly planned out and were literally given just because I thought that is what math teachers did.

Enter Dr. Craig Schroeder in my life, first as my math coach and later as a colleague who began transforming my views on homework.  I really couldn’t argue with his no more than four homework problem philosophy as he explained that kids either knew how to do the work after the first four or did the first four wrong and every problem after that wrong all while making the misconceptions in their mind more permanent with each problem.  He also added that even worse is when the homework causes kids to hate math even more as they struggle with their 28th problem of the night.  I began to model my homework after his, giving three to five homework problems a night and making sure they were problems that were really worth my students’ time and effort.  However my 1st grader is making me question even that policy.

My experience on the parental side of homework has been interesting.  For the first time I see the time constraints that kids and parents are up against with homework.  Most nights by the time I get home from work, we eat dinner, bathe the kids, and get homework done it is straight to bed for our son with little time to no time for playing or family time.  Of course that is on a normal night, on nights when there is soccer practice or an event at his school we end up pushing bedtime back sometimes by an hour or more to make sure we get the homework done.  It really isn’t that his teacher gives an abundance of homework but by the time we get through the spelling, site words, and nightly reading it seems as though our time is up.  I will give his teacher a tremendous amount of credit for sending home math games weekly instead of the usual worksheet as they do provide us with an activity that is fun and educational.

Although we do spend a great deal of time on homework as a family that is not what has caused me to shift my view points further on homework.  Instead my thoughts have shifted as I sit and work with my son nightly helping him finish his tasks while thinking about all the students who don’t have a parent at home to support them.  So many of my students have parents who work in the evenings, who have other responsibilities, or who simply don’t make homework time a priority.  Homework must be such a struggle for those kids.  And what about the kids who are over committed due to sports, clubs, or even extra responsibilities at home?  I have received e-mails from kids asking homework questions well past 11:00 at night and when I asked the kids while they were up working on homework they very honestly say it is the first chance they have had to get started.

More and more I am starting to see that maybe homework doesn’t have to be a required part of math class.  I am currently still giving those short three to five problem homework assignments but am beginning to work through other ideas in my head of a math class with no required homework but maybe just suggested problems to help check for understanding that students could do by choice.  I don’t know what the answer is yet but I know there must be a better way for all students.

Warm-Ups Transformed My Classroom

I have always had a love/hate relationship with warm-ups and flashbacks.  At the school where I teach they are required and I truly to get the “why” behind using them to start class.  After all, I honestly need that 5 minutes to take attendance, answer questions, and deal with whatever 7th grade crisis just transpired in the hall.  However, I have always struggled with what that warm-up or flashback should look like.  So I have done what many teachers do, that good old skill and drill warm-up.  You know the process, you give the kids the five questions, they pretend to do them, you go over the answers, they write the answers down and pretend they got them all correct.  Great learning going on there right?  (Side note, sometimes I think about the ways I have taught kids in the past and cry a little on the inside)

Time to Change

Enter this year’s teacher led TMC conference.  No I didn’t attend the conference but I felt like I did.  I anxiously awaited every tweet, read every blog post that came out of it, and resolved that even if I wasn’t there I could certainly still use it to make myself a better teacher.  Of course, I quickly became overwhelmed with so many amazing ideas at once so I decided I needed to focus my efforts and energies so I started working on my warm-up dilemma.  I started by reading this blog post and then that quickly led to others and as I read post after post about teachers who had leveraged their warm-ups in the classroom to really improve student learning.  I knew this change was needed for me and was doable so I created this Warm-Up to use this year in my class.

Each day we do Estimation 180.  I know some only incorporate it once or twice a week but due to the fact that I love it and the kids love it I knew I needed to do it every day.  The kids fill out the hand-out provided on the website and also send their estimate in on their clicker.  This allows me to provide an incentive to our best estimator (using our team money system) and once I display the live results it gives us some great talking points.  We talk a lot about the estimates, what we know was too high or low, why some answers were more popular than others, etc.  Besides just the reasoning and number sense provided by the activity I love the focus that we have been able to place on finding the percent of error.  Percent error is such a big 7th grade Common Core Standard and is so valuable in students being able to reason with percentages.  I have been amazed by the results so far.  In a class that is about 30% English Language Learners and 45% students with disabilities, 92% of students turned in an Estimation 180 sheet for the first 20 days of school that was filled with beautiful reasoning strategies and high quality percent of error work.  I can’t begin to tell you how rare it is for students to put that much effort into a warm-up sheet.  And to date, 84% of students in that class have currently mastered finding the percent of error with no formal instruction only the focus we have placed on it during our Estimation 180 time.


The rest of our warm-up time changes based on the day of the week as follows:

Math Talk Monday

Counting Circle Tuesday (There are tons of great resources out there for this, just google Counting Circle!)

Would You Rather Wednesday

Tough Pattern Thursday 

Find the Flub Friday  (I just write a problem on the board and purposefully work it out incorrectly.)


I love the focus this has allowed us to place on mathematical reasoning and processing and not skill and drill.  I love that the kids have a few minutes to share their ideas and just talk about math.  I love that kids the used to pretend to do their warm-ups and then just wrote down the answers have bought in and work diligently so that they have something to share with the class.  I love that we are focusing less on the right answer and more on the right reason.  I love that when I read their warm-ups at the end of the week that I can see the effort they have put in.  I love that warm-ups have went from my least favorite part of class to the most valuable time we spend.

Classifying Rational Numbers

I feel as though teaching classifying Rational Numbers to my 7th graders is probably not the most exciting lesson of their mathematics career.  There is no awesome Mathalicious lesson on it and most of the resources I have found are pretty dry and boring.  This is the story of how I developed an understanding of Rational Numbers without falling asleep.  If you have a suggestion on how to make it better I would love that!  I always struggle with this standard and making it come alive for kids.  I do however feel it is an important place to start in 7th grade.  Before we can really start to work with integers it seems logical to me that they must first understand different types of numbers and what integers really are.

The Hook

I always start the lesson with a blank venn diagram of the Real Number System on the board.  I have the kids call out numbers while I place them in the correct category.  The goal of the game is to get the students to figure out the characteristics of each group of numbers.  I provide rewards each time a student figures out a classification.  Generally the students repeatedly call out natural numbers and get frustrated when only the inner most circle is filled with numbers.  After a few minutes they usually branch out into the whole and integer categories but that elusive outer ring usually leaves them baffled.


It is always a major breakthrough when someone calls out that first rational numbers and then the kids start to pick up on the system more quickly.


And then once someone throws Pi into the mix if really starts to take shape:


Eventually after a lot of discussions the kids usually come up with descriptions for each category and I help add the vocabulary in.  




The Follow-Up

We followed this up with a sorting activity.  The students drew the Real Number System diagram on their tables and I provided cards with different numbers on them.  Students then worked in their groups to place each number in the correct category.  I provided feedback to groups as needed and asked guiding questions to help tables who were struggling.



As a separate follow-up activity we also completed the Is It Rational FAL activity found here.  What I love about that activity is that it really helps kids focus in on looking for the pattern in what really makes a number rational or irrational.  I feel like too many kids have the misconception that rational numbers = fractions when the definition is actually a ratio of two integers.  There are plenty of examples of things that look like fractions but are not actually rational numbers and it is important kids can recognize the differences.

Although the students really seemed to grasp on to the concepts of classifying numbers I am always looking for ways to improve.  How do you teach classifying rational numbers?


If You Build It They Will Come

I have no idea why but 8 days into a new school year and I feel like this quote from Field of Dreams best describes what we have going on.

My collaborating teacher and I knew that we would have our work cut out for us this year teaching two collaborative classes. Both classes are filled with students who hate math, students who have been taught that they can’t do math and believe they aren’t as smart as the other students in the school. Although I realize we still have a long way to go it has been amazing to watch their mindsets change in the few days we have spent with them.

 How to Learn Math for Students

We made the choice over the summer to have the kids go through Jo Boaler’s How to Learn Math for Students course. WOW. I can’t say enough about this course. The kids love the videos and they are really buying into what Jo is saying about math ability and growth mindsets. Their reflections on the videos have been beautiful to read, just a few of the comments thus far have been:

“I never knew everyone could be good at math”

“I know I am not bad at math I just had a bad experience”

“Anyone can learn math”


And those are just a handful out of many more like them. I am so excited to see how their growth mindset continues to develop as a result of this course.

Skill and Drill No More 

Equally as important has been a shift in our classroom warm-ups thanks to the MTBOS. I will not lie I was one of those terrible drill and skill warm-up teachers before this year. However I spent all summer pouring through blogs and reading everything that came out of the TMC 14 conference to figure out a better way to engage kids in math. I settled on Estimation 180, Math Talks, Counting Circles, Would You Rather, Visual Patterns and Find the Mistake along with a weekly reflection.   This one shift has probably had the biggest impact on the class. From the moment they enter my students are engaged in high level math and having deep discussions that we rarely had before.  I will make another post just on the warm-ups later!


What I Know

We are eight days in and I can’t guarantee you that every child will master every 7th grade standard or that every child will leave as a mathematician. Here is what I do know though:

Kids that have never enjoyed math come into the room smiling

Kids that said on the first day that math was their least favorite subject wrote on their first weekly reflection that is was their favorite class

Kids that always failed every quiz passed our first one with a B or better.


Maybe it has just been 8 days but it has been the most beautiful 8 days of my teaching career.

Classroom Remodel

The Wall

I am not by any means the craftiest, neatest, most organized, or creative teacher in our building but I still decided to take on the project of redoing my classroom decor this summer.  My biggest problem has always been my “wall”. My classroom has a removable wall that is made of some sort of upholstery like fabric that drives me crazy because it is dull an dingy looking.  The first year I was there I did nothing to it and years  two and three I covered it with fabric but I was tired of looking at that as well.  The biggest problem with the wall is that we have to be able to open it for team incentives, open house, etc so I knew I had to be able to cover the panels individually to keep the functionality.

The "Wall" before the redo

The Wall Before the Redo

Enter Pinterest

I found this pin with lots of different uses for scrapbook paper but I knew I would have to toughen it up to get it to last for 177 days with 120+ kids.  I found a scrapbook paper pad that I liked that had 20 different patterns with 4 sheets of each pattern.  It took 2 to do the wall and and another one to finish up some other stuff I wanted to cover.  Purchased on sale at Hobby Lobby the wall cost right at $15.00 and the whole room was $21.00…way cheaper than the fabric I had purchased in years prior!

I laminated each sheet individually and then cut them out (yes there was lamination scraps all over my house!).  Next I just pieced them on the wall and secured with a staple gun.  I did have to trim some of them so that they wouldn’t keep the wall from opening but in the end it still turned out great. I also made my bulletin board match and hot glued extra sheets to the sides of my book cases and filing cabinets to add to the effect and cover up some ugliness!  

The new wall

The New Wall

I also found some great storage containers to match for  $1 at Dollar Tree and was so grateful to Sarah’s blog for the FABULOUS classroom posters.  They were the perfect finishing touch to my room.  

Matching bulletin board


All in all, my entire remodel was less than $30.00 and I am thrilled with the results.  I just wish I had taken more before pictures to show!

7th Grade Common Core Taboo Cards

One of our MDC PD days last year included a math vocabulary taboo game that was really fun. I printed the cards out to play with my 7th graders but found myself removing lots of the cards that were high school vocabulary based and that my students didn’t know yet. I developed these cards by 7th grade Common Core Unit with the thought that I would print each unit on a different color paper so I could quickly identify what unit the vocab came from. If you can use these in your class please do! We always have days were we have assemblies and come back to class with five minutes left and need an activity or the power is out and we need a back up activity or we got done with a test early and need an activity so I am hopeful these will help make good use of that time.

Also, after you have made taboo cards for rate, unit rate, constant rate, proportionality, etc the taboo words begin to run together so if you have a suggestion to make one of the cards better please let me know!

Expressions and Equations Taboo

Geometry Taboo

Number System Taboo

Ratios and Proportions Taboo

Stats & Prob Taboo