What Did You Say About Cheeseburgers

Standard

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’s Inevitable 

Standard

Let’s face it…students coming to us using k-c-f (invert and multiply) to solve division of fraction problems is inevitable. There are too many teachers who rely upon this method as the sole method for dividing fractions. From my experience with teachers, many of them believe you cannot divide fractions, you can only multiply them, which makes invert and multiply a logical method to use. 

Does this mean you should continue on with this trend of misunderstandings when students use this within your classroom?  If you’re teaching for correct answers only, I’m sure there was a resounding “yes!” to my question. However, if you teach for understanding which allows students to build upon their knowledge, apply concepts to new information and see patterns and structure you probably paused to ponder what other methods are there. 

The black hole of resources within nzmaths (Nzmaths: Lesson on Dividing Fractions) houses a wonderful set of lessons that help guide students through developing an understanding of dividing fractions. Towards the end of the sessions students are encouraged to look for patterns when dividing whole numbers by fractions. 

  
Two things I added when implementing these sessions with students were beginning with 1 whole and asking students to create a visual representation. 

   
 

Because many teachers do not conceptually understand the invert and multiply method, I will not rob you of an opportunity to make sense of it for yourself. Try it out and leave a comment.