Deep Procedural Fluency with Long Division

Imagine you are a 4th, 5th, or 6th grade teacher (perhaps you don’t need to imagine this).  You have just embarked on a unit that addresses division standards (4.NBT.6, 5.NBT.6, or 6.NS.2).  A parent demands that you challenge his/her child because the student has already “mastered” division.  True or not, what do you do?

If this were a game of Family Feud, Survey Says #1: “Give the student bigger numbers to divide!”  Survey Says #2: “Put that kid on Khan Academy during math class.”  And Survey Says #3: “Go ahead and teach the next grade level’s division standards.”

survey

Good thing this ain’t Family Feud.

In my quest to find a simple but rangy task that emphasizes deep procedural fluency, I constructed a “missing digits division” challenge.  I finally got a chance to test it out on real students in a real classroom this week, and the outcome was pretty darn cool IMO.

So, I started by simply taking a division computation problem from a math textbook.  SPOILER ALERT (stop reading here if you desire and go to the next paragraph).  The problem was 1,971 ÷ 27.

I solved the problem using the standard U.S. algorithm (although the way I represented the problem on the student task sheet is a hybrid of both the standard algorithm and partial quotients).  Then, I erased several of the digits in the dividend, as well as many of the digits in the partial products (generated by multiplying the partial quotients and divisor).  Put a box in the place of every missing digit, and voila!

missing-digits

I gave this task to a group of eager 5th graders with no scaffolding or assistance whatsoever.  Like most great tasks, you just let them dive straight in together and find their own entry points.

file_000

What happened next was intriguing.  Several students started working backwards and about half of them wrote 91 in the boxes at the bottom of the problem.  They reasoned that, based on the first subtraction step, 91 could be the only possible difference.  But, others argued that 91 wasn’t possible because it isn’t a multiple of 27.  But 81 is.  But the group remained divided (pun intended).file_001

After wrestling with the problem and talking it out together with partners, several students reached a dividend of 1,171 (because it is 81 more than 1,090).  Again, these students were exclusively focused on working backwards, so they were paying full attention to the subtraction part of the division procedure.  This led every one of the students to a quotient of 43.

file_003

Excitedly, they looked at me and asked me if this was the correct solution.

Poker face.

“Do you know the answer to this problem?” one of them asked, innocently.

“I used to,” I responded.  “But I forgot.”

They looked at each other like, “SERIOUSLY?”  Their faces seemed to say, “Dude, you created this problem, right?  And you are a teacher, no?”

Even though I was dead serious (I did forget the solution), I did not want them to look to me to evaluate their ideas – it is much more powerful when they own the math themselves.  I should not be the arbiter of truth – they should.

“We could use multiplication to check,” one student said.  So, they did.  And then they realized that the product of 43 and 27 was not the same as the dividend.

After some discussion, one of the students reasoned, “Does this problem have a correct answer?”  Was he implying that I botched this task and made a typo?  “That’s a great question,” I responded.  “Maybe I did screw this up somehow.  Or… maybe you all did.”  They smiled and several of them started erasing and feverishly searching for a correct solution.  This is SMP 1(perseverance) at its best.

These students knew that the units digit in the quotient (i.e., 3) was correct, so they experimented with changing the tens digit.  Within moments, they realized it had to be a 7.  The product of 73 and 27 is 1,971, which finally made everything else fall into place and make sense!

file_002

Their followup task was to create their own missing digits division problem and share/trade with a friend.  Man, did they love doing that!  They couldn’t wait to try to stump a buddy.

So, if you are looking for an easy way to challenge students to explore division procedures at a deeper level, give this a try and let me know how it goes.

Oh yeah — and for kicks and giggles, if you’re looking for another challenging division problem, try this problem I recently submitted to openmiddle.com for review.

 

Fill in the blanks using all different non-zero digits (except the numbers 1 and 4, which have already been used) to make the greatest possible quotient.  Hint: There is NO remainder.  What number should ideally be placed in the hundreds place of the quotient?  Why?

openmiddledivprob

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s