Numberless Word Problem 1: Division in the 3rd Grade

I had the pleasure of working with 2 amazing 3rd grade teachers last week.  They had the usual concerns and frustrations about teaching word problems and were eager to find a better way to get their kids enthused and engaged.  So, we chose to do something different.  And I’d like to tell you about it.

About a week or two ago, I came across Brian Bushart’s (@bstockus) posts about using/writing Numberless Word Problems (NWPs) that you can read about here.  (Regina Payne (@reginarocks) has also been involved in launching the work around NWPs as well.)

I showed Brian’s post to the teachers.  After a quick read, they were hooked and we set to work.  (Reason 1 why NWPs are awesome:  They trigger a “I can do that!” response for teachers.)

We dove into their curriculum and found the usual dry word problem that sometimes leads off a lesson:  Yasmin has 24 large apples and she makes 8 loaves of apple bread.  Each loaf contains the same number of apples.  How many apples did Yazmin put into each loaf?

The textbook heavily scaffolded the solution process with the usual stuff that sucks the joy out of learning and exploring.

So this is what we did and how we did it.

The first thing we did was change “bread” to “pie”.  Partly because pie is more delicious and also partly because we thought the loaf/loaves represented an unnecessary hurdle for English language learners.

Then the teachers decided to help bring in some roleplay and story telling to set the scene and frame the work for the day by displaying the following slide.

Screen Shot 2016-03-08 at 11.10.45 AM

Then they showed “Clue 1”.

Screen Shot 2016-03-08 at 11.13.56 AM

They asked:  “What math do you see in this sentence?”  Students paused and thought and then started talking with their tablemates.  All students were engaged and discussing what “several” and “some” meant.  One student ponders aloud, “How many apples does she have?”  (Reason 2 why NWPs are awesome:  Open questions mean all students can play.)

Student responses were then charted.  One student mentioned that large apples means less apples were needed than if she used smaller apples.  Classmates listen and nod in agreement.  Another noted that “some” pies meant that there had to be more than one pie.   (See image below for a capture of their responses.)

On to Clue 2:

Screen Shot 2016-03-08 at 11.24.41 AM

Teachers asked:  “How does this new clue change what you already now?”  Full discussion starts to unfold at their tables.  Some students hollered out:  “It’s a multiplication problem!”  “No!  It’s a division problem!”  We asked for evidence and they talked about counting equal groups or making equal groups.  Another student suggested that there had to be more apples than pies which led to an interesting discussion.  Some students wanted to know how the apples were sliced.  (Reason 3 why NWPs are awesome:  Students drive the conversation!)

Clue 3:

Screen Shot 2016-03-08 at 11.30.03 AM

Teachers asked:  “What do you notice?”  Then after student conversation:  “What do you wonder now?”  Student responses…  How may pies did she make?  How many pies can you make with 24 apples?  How many applies will she use for each pie?  I bet she made 6 pies with 4 apples in each pie.

Teachers also asked students to make an estimate.  “How many pies do you think she made with 24 apples?  And why?”  (Reason 4 why NWPs are awesome:  Students can make estimations to their own questions.)

Finally, the 4th Clue:

Screen Shot 2016-03-08 at 11.43.51 AM

Teachers:  “What questions can we answer now?”  More student discussion and exploration.  How many apples will she use for each pie?  How many slices of apple for each apple?  each pie?  How many slices of apples are in 8 pies?

Students set to work answering their questions.  A few interesting things to note:

After 45 minutes, all students are still into the problem.  Butts are out of seats.  Heads are together.  Students talking to students.  Introverted students scratching their heads as they silently look over their own work.

Two stronger students started wondering how many apples would she need for 10 pies?  They started to list some ideas and started to make the rudimentary beginnings of a ratio table and informally talking about 3 apples a pie as a rate!  I was blown away.

And then…. the lunch bell and students groaned!  (Reason 5 why NWPs are awesome:  Students are bummed out by the lunch bell.)

Closing thoughts:

  1. I can’t wait to do this again.  Neither can the teachers or the students.  They want to do it weekly.  Creating an NWP was simple and easy, and they were able to create them using their existing text.
  2. At one point during the lesson, the teacher asked a “closed” question during the lesson.  The shift in the energy in the room was drastic and sudden:  students got quiet, discussion got stifled, and the teacher became the only math-doer in the classroom for that moment.  The students were afraid of giving the wrong answer to a right/wrong closed question.  The teacher realized the shift in energy; we (the observers) noticed it too.  After a moment, the teacher backtracked and said “Let’s come back to that question later, OK.  Who else has something interesting to say about this problem?”  BAM!  Students are back on track and thinking and exploring and doing.
  3. But that moment was my key piece of learning.  Open questions allow for all students to play.  It demands that stronger students be more than algorithm followers and actually do math.  It makes discussion and sharing of ideas easier.  It reduces anxiety. Obviously, closed questions are necessary too!  But if dialogue, discussion, and full student engagement are goals to target while tackling word problems, then open questions are the way to go!  And NWPs allow for that to happen in a structured, scaffolded approach that teachers can use to guide learning.
  4. I’ll never look at a math textbook the same way again.  I’m looking forward to seeing and hearing about what this looks like in middle school classrooms and in high school.

So:  What do you think?  How could you make it better?  What learning has this created for you?


Two remaining tidbits because I’m long-winded like that.

Here are the student charts if they interest you.

Screen Shot 2016-03-08 at 11.32.27 AM


Here is a cool extension that they created to help reinforce learning.  They created a clue booklet for another NWP.  The goal was that students would be able to write and show they’re thinking at each step of the way.  I tried to show that in this video below so you could see the problem, but it’s a comical disaster.  Who knew that stapled paper was so hard to work with??  Anyway, you might find it useful.

2 thoughts on “Numberless Word Problem 1: Division in the 3rd Grade

  1. Thank you for documenting this experience with numberless word problems! I especially liked the way you modified the problem in consideration of ELLS. I also liked your honest commentary throughout of what was awesome and what was not so awesome. This was such a treat to read. Thank you for sharing!

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