Decimal Division Strategies and Sense Making for 5th Grade

It’s my hope that there are other 5th and 6th grade teachers out there that might find this analysis useful if they want to use this lesson in their classrooms to engage their students in exploring decimal division (5.NBT.6, 5.NBT.7, and 6.NS.3).  Please feel free to plagiarize and make it your own as you see fit.  I’d love to hear from you if you use it so we can get better together.  (Special thanks to Graham Fletcher for providing such an array of amazing 3-Act Lessons!)

Before reading further, it may be useful to watch the lesson resources on Graham’s website.  Go ahead and check them out here.  I’ll wait.

And…welcome back!  Let’s dive in.

Some quick background:

I’ve conducted lesson inquiries using Tomato-Tomato with two teams of 5th grade teachers at two different schools.  In total, I’ve taught, co-taught, or observed this lesson five times.  Every lesson was a unique experience with it’s own twists and turns and choices.  The lesson pathway below shows the main flow of the learning experience and took about 60 minutes.  Further analysis and takeaways can found at the end of this post, but I’ve tried to embed a lot of our thinking within the flow of the lesson.

This lesson was taught to 5th grade students that had not yet explored decimal division operations.  I think Tomato-Tomato is a more valuable learning experience for students this way because they are encouraged to use a wide variety of sense making strategies.  It also creates the headache (the need) for the aspirin (the long division algorithm with decimals).  (Find out more about Dan Meyer’s headache/aspirin metaphor.)

Our objectives (as teachers):

• Students will be motivated and inspired to ask questions, engage in discourse, and persevere in solving a task.
• Students will show what skills they have using models or symbolic operations to solve a problem.
• Teacher will foster mathematical inquiry for groups using questions and discussing problem solving strategies.  Teacher will use differentiated versions of the problem if necessary for struggling students.
• Teacher will monitor student learning, look for anticipated students responses, select and sequence student work to share in discussion, and make connections between their models and symbolic notation.

Specifically, we wanted to see what sense making strategies they choose.  Would they draw models?  Which models?  Would they do repeated addition or subtraction?  Trial and error with multiplication?  How many would attempt the long-division algorithm?

We were also asking these questions as observers:  Are students enjoying math?  What mood structures their interactions with each other?  Where are they emotionally?  What’s their level of initiative?  How do they self-advocate?  Are they going to be able to do this without my direct guidance?

Learning objectives (for students):

• I will show how I can use math strategies to solve a problem.
• I will collaborate and actively listen to my classmates.

Some quick notes here…  We wanted to keep the language very simple.  We also felt strongly about using the general phrase “math strategies” instead of talking specifically about the four operations or drawing models.  We certainly didn’t want to use the word division.  We wanted to keep the process as open as possible for a variety of pathways to the solution so that all students could play.

We added the second objective because if we wanted to see students collaborating in sharing strategies and engaging in discourse, we needed to make that explicit with our students.

These objectives were posted clearly on the board and referenced throughout the lesson at transition points.

Into:

After explaining the learning objectives to the students, we launched by getting students to think about ketchup containers.

Today we’re gonna talk about ketchup.  Yup.  Ketchup.  You know that ketchup comes in different sizes.  At McDs, it comes in those little packets you can’t open easily.  At your house, it probably comes in bigger bottles.  At Costco it comes in giant tubs.

At this point, kids are nodding in agreement.

We’re going to watch a short video.  I’m really curious about what you see in this video and what you notice and wonder.

We played the short Act 1 video showing both bottles of ketchup.  After the video, we did a little pair/share before launching in to a Notice-Wonder exploration.

The Notice-Wonder strategy is a very effective discussion strategy to get all students talking (because everyone can describe something they saw and ask an interesting question regardless of their math ability).  It also lowers the floor into a math task because it’s a low-risk strategy that allows student discourse to describe the problem rather than having us (the teacher) “give” them the problem.

We paused the movie about 2 seconds before the end of the video so that the image remained on the screen.

Tell me all the things you notice.

We did not scribe these statements.  We did often ask:  How many of you also said the same thing?  Who can say it in a different way?

We used these questions when students shared “noticings” that were about the sizes of the bottles because we wanted to make sure that this concept was clear to all students.  These questions allowed for more student participation and provided opportunities to practice listening to each other.

Now, I want you to ask some questions about this video.  What questions can you ask?  What do you wonder?  

In some lessons, we didn’t need to ask the wondering question because we were able to paraphrase the things they noticed in way that led to the following key question:

How many of the little bottles ketchup make up the big bottle of ketchup? 

We scribed this question clearly on the board below the objectives.  We had them describe different ways to ask the same question because we really wanted to make sure that ALL students understood this question conceptually.  How many little bottles could the big bottle fill up?  How many little bottles go in to the big bottle?  How many times bigger is the bigger bottle?  How many times smaller is the smaller bottle?

We also unpacked what we meant by “filling” bottles.  There is a gap of air in each bottle.  In other words, we were talking about the volume of the ketchup in the bottle, not the volume of the bottle itself.

With our key question on the board, we continued:  Huh.  I’m curious what you think.  Can you make estimate?  Students make an estimate.  Then there’s a pair-share moment.  We had them write their estimate on their paper to hold them accountable.

Who would like to share an estimate?  Did anyone get fewer?  Greater?

We scribed these estimates on the board, loosely ordering them from least to greatest.  In some classes, we attached names to estimates.

You have made some interesting estimates.  As you know, estimates are just educated guesses.  If you wanted to make your estimate more accurate, what more information would you like to have about the ketchup bottles?  (And why is that information helpful to you?)

This phrasing really sets up the pivot between Act 1 and Act 2 of the lesson.  It hinges their question to a reasonable numeric answer.  It also makes it explicit that math is a device that can allow us to make our reasonable estimates more accurate.  And that sometimes, math only leads to more reasonable estimate, not an exact answer.  (This lesson and Array-Bow do a particularly job of showing how math can be used to model situations that yield reasonable, but not exact answers.)

For these reasons, I often use this language at this transition point in many 3-Act math lessons.

In every lesson experience, students realized that they needed to know how much ketchup is in each bottle and asked for that information.  (In some lessons, students wanted to know the size each bottle.  We usually let that question stay even though it is slightly inaccurate.  In some cases, we pushed back a bit more and asked students to clarify that we were talking the volume of the ketchup, not the volume of the bottle.  And while we’re at it, we’re technically talking about the mass of the ketchup and not about volume at all…but…)  In one lesson, we had to nudge them there with some effective paraphrasing of student responses.

Through (part 1):

Because we wanted to focus on decimal sense making strategies (and also because the numbers are smaller and therefore more accessible to all learners), we did not use Graham’s Act 2 image.  Instead we wrote the sizes (14 oz and 2.25 oz) on a frozen image at the end of the movie.  We did not write the weight in grams.

I’m curious to see how you approach this problem and how you make sense of it.  Let’s do some math and see who got the closest estimate!  

This is that scary moment in the lesson.  Will they take off?  Will there be launch failure?  Have we baited the hook with a juicy enough worm?

In our preparation for this lesson, we talked a lot about this moment and what our choices were as a teacher.  If we wanted students to persevere and make sense of the problem, we knew we needed to back off as instructors and resist the temptation to intervene with direct instruction.  Prior to the lesson, a few teachers identified a handful students that might need a quick check-in/pep talk/accommodation right from the start, but for the most part, we had made a commitment that for at least the first 5-10 minutes, we would just float and observe and intervene only with questions and encouragement.

After 5-10 minutes, the teacher would step back and see which students were not succeeding/engaged and determine if it was because of a lack of clarity, a lack of skill, or a lack of confidence/desire.  In most classes, there were usually 2-5 students that needed this boost.

Some quick notes here:

• Encourage students to come up to the board.  Because then this happens:

• “What if the small bottle was 2 oz?  What would your answer be?” was a question we phrased to struggling students who needed a lower floor.
• A lot of students immediately multiplied 14 x 2.25.  In some classes, the reasoning error was so prevalent that we called a quick timeout and addressed it as a whole group.  We compared the product (31.5) to their estimates.  This helped them to see that their answer didn’t make reasonable numeric sense.  I also started to draw a diagram of several 2.25 oz bottles on the board to show them that the product could be the total ounces in fourteen 2.25 oz bottles.  This image helped to show that multiplying 14 x 2.25 didn’t make conceptual sense for this problem.
• We made an intentional effort to praise students explicitly for sharing their thinking and reasoning with each other.  We wanted the focus to be off of just answer getting because we wanted to see of lot of this:
• Since we also wanted to focus on multiple strategies, we encouraged students to show their thinking in as many ways as they could.

Through (part 2):

The numerical answer to the question is 6 and 2/9 or 6.222….  Most students quickly surmise that the answer is in between 6 and 7.  Much of the amazing joy of this lesson is watching how students grapple with and make sense of the amount of the remainder.  It was a big part of our discussion at the end of Act 2.

I want to unpack some student work samples because I think they are important to sequence from most conceptual (visual diagrams) to most abstract (long-division algorithm).  We anticipated many of these approaches and made note of which work samples we wanted to select for discussion at the end of Act 2.

Students made pictures.  We encouraged these students to put numbers to their bottles.  They quickly realized they needed more bottles; 6.75 wasn’t enough.

Students made a tape diagram.  We liked this example because it linked to the diagrams like the one above.  It also helps students see that 6 bottles (13.50 ounces) is too few and that 7 bottles (15.75 ounces) is too many.

The tape diagram also sets up nicely with student work samples that showed repeated addition and multiplication.

We hoped to see students using repeated subtraction from 14, but only a few students did that.  We tried to leverage this work as much as possible because it creates the conceptual need for division as an appropriate strategy here.

Many students set up the problem for the long division algorithm, but quickly realized that they didn’t know how to handle the decimal in the divisor and abandoned their plan.  This created the headache for the aspirin (decimal division) that they were going to learn in the next chapter.  We stressed this very explicitly with students to help them be aware that they’ve recognized a need for what they were going to learn about tomorrow.

Some students who were confident with the long division algorithm persevered onward.  This student was even able to explain why he needed to multiply both dividend and divisor by 100.  None of us knew when he learned this and how he came to know it.  

This student persevered through some of the algorithm, but interestingly enough, she struggled to explain what the two-tenths meant in the context of the problem.  In other words, she knew that 14 ÷ 2.25 was close to 6.2, but couldn’t think in parts of bottles and her work served as an interesting display of procedural fluency without conceptual understanding.

After about 20 minutes, many students were starting to run thin on productive struggle.  Most had figured out that it was between 6 and 7 bottles and could show why in several ways, but we found lot of hidden value in going deeper by asking students:

Do you think the answer is closer to 6 or 7?  How do you know?  

For many students, setting up a number line like this was helpful and offered an opportunity to bring in this model to the mathematical process.

While this problem is set up beautifully for double number lines and some 6th grade proportional reasoning skills, we went with a simple number line like this.  Like many students, this student realize that he “was 0.50 ounces short” and “that isn’t very much” so that the answer had to be closer to six bottles because “0.50 is way less than half of 2.25.”  Several say the answer is closer to 6 because 14 is closer to 13.50 and 15.75.  A few strive to get more exact with a variety of reasoning strategy.  The conversation around this topic was vibrant.

With this number line in place, a lot of students were able to grapple with the remainder issue in their own way.

A quick aside:  a colleague witnessed one student think hard about this problem.  He wrote the number 6 down before he changed it to 6.5 after another of moment of thought.  Then he thought longer and wrote in 6.25.  He listened to others, but never wrote anything down or showed calculations.  We asked him to share his thinking, and he willingly attempted to explain, but really struggled to articulate why he knew the answer had to be less than 6.5.  “Because that’s how the numbers are” was his ultimate answer.  It was an interesting occurrence where conceptual reasoning and number sense skills can precede that ability to communicate reasoning.

Beyond:

When the ceiling of productive struggle had been reached for most students, we selected examples of the student work we wanted to share and sequenced them similar to as they are presented above.  At certain points, we had students share their work to the class.  We made a point to start with conceptual solutions first (visual models), then solutions using addition/subtraction notation, then solutions using multiplication/division, and then discussed strategies for figuring out the “closer to 6 or 7?” question.

After we had shared our thinking, we concluded that as a group we were sure that our answer was between 6 and 7 and that is was closer to 6 and might be close to 6.2 or 6.25.  We then compared that to class estimates from earlier.  We talked about the value of mathematics because everyone was able to use numerical reasoning to move their original estimate closer to the real value.

Want to see the answer?

We are about 45-50 minutes from the beginning of the lesson, and kids are still jazzed at this point and all are leaning in.  Before playing the video we explain that they’ll watch colored water instead of ketchup because ketchup is gooey and it’s hard to get it all out.

We then watched Act 3.  After the 5th bottle, we pause the video and students groan with unfulfilled anticipation.  But it’s a teachable moment.  We now have more information.  How do we feel about our math answer?  It’s a good chance to keep the answer (6.2) contextual.

We then play the video through and pause with the final reveal on the screen.  The video provides a tremendous opportunity to reflect.  For starters, it shifts the answer key away from the teacher.  It sits there as an objective fact and exists as raw evidence of the answer.  It also allows all students to see that they had math skills that allowed them to improve the accuracy of their original estimate.  Lastly, it creates the intellectual need to learn some more skills about dividing with decimals.

We concluded Act 3 with these reflection questions.  Students were asked to pick one and write a response on an exit ticket or in their journals.  After 5 minutes of writing, we asked for student volunteers to share what they wrote.

 

Reflections, Takeaways, Analysis:

• This activity is naturally open and differentiated, it appeals to all learners regardless of mathematical skill.
• Likewise, it allows for a variety of learning styles to shine and diversifies our idea of what math talent looks like.
• It’s a great launch in to decimal division.
• It is fertile ground for student discourse, discussion, and perseverance.
• Estimation and Notice/Wonder are powerful instructional tools.
• Sharing student work to frame discussion creates a more authentic context for direct instruction about mathematical structures, strategies, and reasoning.
• We should have listed the student strategies on the board.  This would have strengthened their learning experience.
• If we want to celebrate mistakes, we should encourage students to stop erasing their work.
• What are some upcoming concepts in our curriculum that we can open up with this 3-Act math lesson structure?
• These types of lessons can be a rewarding, whole class collaborative assessment.  How close can we get as a class?  Let’s see what we can do!

As always, feedback and comments are welcome.  What inspired you?  What opportunities did we miss?