What Did You Say About Cheeseburgers


Yes, it’s how we learned it.  Yes, most people will say, “It worked for me, so it will work for them!”  The standard algorithm has definitely been a bone of contention in many conversations.  On another, but connected note, while watching Spiderman into the Spider Verse, my youngest was singing along with one of the songs.  A friend of mine made the comment, “Kids can hear a song once and be able to sing it, but yet they can’t remember things in school.”  (With the voice of Carrie) I couldn’t help but wonder, “If concepts had more structure that is easily identified by students, would learning be more fluid”.  This reminded me of a talk I watched of William McCallum on The Story of Algebra.  In it he discussed finding structure within a poem to help memorize the poem.  He then connected it to the structure found within the Common Core Standards.

In 5th grade, students solve division of whole numbers using partial quotients.  A huge component of partial quotients is considering the entire quantity of the dividend and the application of place value understanding.  For example:

Partial quotients example.001

When dividing 691, students should think how many 8s are in all of 691.  After working with multiple of 10s in prior opportunities, students will conclude there are about 80 eights in all of 691.  Fast forward to 6th grade when students are first introduced to the standard algorithm for division, it’s important to carry that same thinking with a heavy undertone of place value understanding.

In our first session as Great Girl Mathematicians, we reviewed division using partial quotients.  By the second session, a young rising 6th grade male joined the group, thus removing Girl from our group name.  The second session was spent connecting partial quotients to standard algorithm.

Partial quotients example.002

We began our session with the partial quotients strategy, but instead of writing the quotients to the side, we staked them on top as they would see them within the standard algorithm.  They could also see the place value connection of the why behind the standard algorithm.  I paralleled this approach with the standard algorithm.  I explicitly modeled my thinking:

“I notice there are 69 tens in 691.  I know 8 x 8 is 64, but I need 64 tens.  Therefore, I can multiply 8 by 8 tens which is 64 tens.  Because it is 8 tens, I will record the 8 in the tens place of the quotient.”

As I did the subtract, I attended to precision by stating, “69 tens minus 64 tens is 5 tens”.  Speaking of it in this manner gives meaning to why the next number is ‘brought down’.  In this problem, the next number is in the ones place, resulting in 51.  Because 8 x 6 is 48, the 6 is placed in the ones place of the quotient.

Partial quotients example.003

As the kids worked through their practice problems, some needed to use the staked partial quotients before rewriting it as the standard algorithm.  Others caught on quickly.   While others grappled with where to place the numbers in the quotients.  Using place value as our guiding thought helped them to make sense of why to record numbers in certain places.

This approach debukes the cheeseburger mnemonic device which causes students to not consider the magnitude of the dividend making the thought “does my answer make sense” seem like a distant after thought.  Being explicit about the place value understanding creates the structure to help students make connections between and among concepts.


It Started as Great Girl Mathematicians


After hearing a friend say how he has been previewing math concepts with his daughter to provide a foundation he knew might be missed in class, I decided to support my daughter in the same manner.  My oldest daughter is entering middle school this upcoming school year.  As a building level coach and district specialist, I’ve observed the struggles students have with transitioning from 5th grade standards to 6th grade standards.  From my experience, this struggle is related to middle school teachers not having a complete understanding of the 5th grade standards and how they directly connect to their grade level standards.

With this knowledge, I decided to create a plan to preview the concepts my daughter will learn in Unit 1 of 6th grade math.  These concepts include:

  • Divide multi-digit whole numbers using the standard algorithm
  • Add, subtract, multiply and divide decimals fluently
  • Apply GCF and LCM
  • Interpret the quotient of fractions

My oldest hasn’t had the easiest time in math.  She has battled with being flexible in her mathematical thinking and gets bogged down with procedures.  In school she has learned many tricks without understanding.  I’ve found it to be difficult to counteract these tricks at home once they are the topic of discussion during class.  But finally, conceptual understanding will have the upper hand, or at least that’s the plan.

As I pondered the best way to approach this preview, I thought of what would be most helpful and effective for Jocelyn.  She’s one who needs to know what’s next and the order in which things will happen.  She’s a social butterfly who is empowered by forward thinking friends.  With all of this, I decided it would be best to open the opportunity up to some of her closest friends.  A group of girls, a mixture of those who have excelled and others who have struggled.

So here’s the plan:

  • Meet at the library twice a week for 30 minutes each time
  • Be explicit about the connection between 5th grade standards and 6th grade standards
  • Assign a “project” for the girls to do for the week while at home.  These projects consist of 3 act tasks, open middle problems and tasks from the Georgia Department of Education mathematics frameworks.
  • Provide the group with notes to use during the unit of study once school starts
  • Use the app GroupMe so the girls can converse about the projects.  I called the group Great Girl Mathematicians, mainly because I wanted to instill that this group of girls is great and are mathematicians no matter the struggles.

In subsequent posts, I’ll share the actual content we discuss.  With them, I hope it will help 6th grade teachers and parents of rising 6th graders to make the transition to middle school more empowering and less awkward.