“We’re Gonna Need a Different Denominator”

Check out this video from Andrew Stadel.  What do you notice?  What do you wonder?

 

I conducted a lesson study about fractions with some 5th grade teachers.  We used Andrew’s elegantly simple lesson called Black Box 2 to get students talking about adding fractions with unlike denominators.  This task is ideal for introducing the intellectual need for finding a common denominator before adding fractions procedurally.  Student discourse is rich and meaningful and lively.  Give it a read.  You won’t be disappointed.

Some background:

I conducted this lesson study with a team of 5th grade teachers.  It was taught to 2 classrooms, and both lessons were unique experiences with its 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 be found at the end of this post, but I’ve tried to embed a lot of our thinking within the flow of the lesson plan.

The content of this lesson centers around adding fractions with unlike denominators, a skill that they had not been formally taught as math students.  Standard 5.NF.1 asks students to “add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.”  This lesson creates the headache, the intellectual need for finding like denominators.

We wanted to collect some data on how students would approach this problem intuitively.  The data would help inform our instructional choices for the upcoming fraction module in the curriculum.

Our objectives (as teachers):

  • The teachers chose the following research question as a way to focus their thinking:  How do we reach the needs of struggling students while also pushing forward and keeping stronger students challenged?
  • We are curious to see if this open lesson format makes math more accessible and meaningful for all students.  How well will this lesson let students “show what they know” and make sense of a new type of problem?

Learning objectives (for students):

  • Today you will make sense of a problem and persevere in solving it.
  • You will talk about your reasoning and listen to others.

Into

To set the mood and energy to achieve our objectives today, we started a Number Talk with Images with students on the carpet.  (If you’re not familiar with the number talk structure, you can check out this video as an example.)  We used the following image and asked students How many do you see?  How do you count them?

We chose this image because we wanted to:

  • Set the tone for student discourse and get them talking right away.
  • Use prior student understanding to review the necessary vocabulary (fraction, one half, one quarter, one whole, etc) instead of having the vocabulary reviewed by us through direct instruction.
  • Show a visual where students were compelled to make wholes out of parts.  This visual gave us a simple, low-floor entry point for students to think about fraction addition.

After some silent think time, we gave them some pair-share time with each other.

We monitored the room to listen to what students were saying to one another.  This data was useful to help us determine what pre-existing knowledge they were bringing to the classroom for the lesson.

Students share their answers.  Common answers heard were 4, 5, 6, and 8.

Well, it surely can’t be all of these answers.  Let’s see if we can use some reasoning to figure out which answer might be best.  Who would like to defend one of these answers?

From there the dialogue moved forward.  We paid attention to student use of vocabulary and highlighted it as it arose.  Notice how the words and fractions for a fourth, a half, and a whole come out as students interact with the diagram.  Here is a common solution:

This student said “I split the halves in half to get fourths.  Then one fourth tucks in to the corner watermelons.  Then the last two halves come together to make a whole.  So that makes 4 for the corners and one more.”  We then unpacked some of this language to make sure that all students had a chance to see the splitting of parts, thinking about the size of parts relative to the whole, and which parts make wholes when put together.  Often times the equations “3/4 + 1/4 = 1” and “1/2 + 1/2 = 1” would end up written on the board.  When possible, we layered words, fractions, and equations on to the visual to help facilitate student thinking.

Did anyone reason it out in a different way?  Not all student solutions are the same.  Check out what this student did:

He said “I made them all fourths and then counted them.  So I see it has 20 fourths.”  Another student shouted out “That’s the same thing in a different way.”  How do you mean?  And let’s everyone listen to see if we agree or disagree or have questions.  

These natural, unprompted student to student discourse moments are quite powerful.  We didn’t predict this solution.  And without encouraging students to think differently and inviting those solutions in to the room, we would have missed out on this nugget of math joy.  Students really loved this solution more because it’s so simple.  (We did too as teachers.)  There’s no need to make wholes.  We can just make fourths and count them.  Brilliant!

We concluded the number talk by asking students to think of the math vocabulary we used in our work today.  We then reviewed words, pictures, numbers to define the terms.

Through:

After returning to desks, students were shown the Black Box 2 video and were asked:  What do you notice?  What do you wonder?  We looped the video so we could keep watching it during their silent think time.  We encouraged students to use the white boards if they wanted to show their thinking.

We spent a lot of time asking students to share what they noticed.  What did the black box appear to be doing?

While students shared, we found it useful to pause the video so students could visualize how 1/2 + 1/2 = 1 and how 1/3 + 1/3 = 2/3.  We went slow here so students could see the connections between the visuals, words, and fractions.  And we were able to make explicit connections back to the watermelon image.  (Note:  I can’t stress enough how valuable it can be to use a number talk with an image that dovetails nicely with a lesson exploration.  All the work we did with the watermelons was still paying off later in the lesson, especially for struggling students.)

We then asked students to share their wonderings.  At this point, most students are clear that the key math question is “What is 1/2 + 1/3?”  Some students used more informal language like “What’s going to come out of the box?” and “Will the green bit fit in the white bit in the orange one?”  Here’s what my board looked like after the end of the notice and wonder launch into the activity.

We printed out copies of the final image.  We wanted to let students physically interact with the visual.  This was crucial to the learning for some students.  They needed to physically see what the new fraction will look like.

We let students tackle this problem individually and collectively.  For many, this was the first time that they encountered adding unlike denominators.  We wanted them to feel a headache.  And they did.

We also wanted to see what students already knew and what intuitive steps they would take.  We monitored the room listening for anticipated (and unanticipated) student responses.  We started to group students by chosen strategies when appropriate and encourage students to collaborate.  We let them work for about 15 minutes before bringing the class to a focus and structuring some discourse.  Here are some samples that we thought were useful to share.

Student Work Sample 1

We saw several students want to make this claim:

For these students, we asked them to cut out their pieces and see if their drawings were correct.  Does the green half fill up the empty part in the orange third?  This was a powerful learning moment for these students.  We pushed their thinking:  So looking at this paper orange and green combination, what might be the value of that fraction?

Student Work Sample 2

Some students intuited that the fraction might be 3/4.  They did this by cutting up the images.  A group came up to the board and drew this image.

Instead of cutting out the blocks, some students used rulers or this technique using a separate sheet a paper to ponder the size of the sum.

In the image above, you can see he’s drawn a line to show that the one half won’t fill the two-thirds.  He also thought that the answer might be 3/4s.

We pushed these students further by asking them:  If you think this colored part is 3/4, what does that make the remaining white part?  “1/4.”  So does that look like 1/4 to you?  Students look at me like I just stole their candy.  “No, the white bit is too small.”  Well, what else might that white part be?  “We’re going to need a different denominator.”  Boom!

This last comment made us smile.  If we’re going to need a different denominator, what might that denominator be?  What would the numerator be?

Student Work Sample 3

Some students wonder if the fraction might be 2/5 by adding the numerator and denominator.  We asked these students to think about the meaning of the denominator and numerator and returned them back to the first few images.  Did your rule work for those?

Student Work Sample 4

Some students wanted to multiply.  We asked these students to think about the previous fractions in the black box.  Did the black box multiply those?  “No.”  How do you know?  “Because a half times a half is a quarter.”  We also asked them to consider the reasonableness of 1/6 as the answer.  It was less than either of the parts and didn’t make sense.

Student Work Sample 5

We noticed that some students had already been exposed to an algorithm.  We challenged these students to show a visual that justifies that their procedure makes sense.  We found that several didn’t not have the conceptual knowledge to back up their procedural skills.

Student Work Sample 6

In each class, about a handful of students reasoned that they could split each of the fractions and make sixths and then figured it that the answer might be 5/6.  Unfortunately, I don’t have pictures of those visuals, but they were useful in the whole class discussion.  But this was the strategy we wanted to highlight most because it shows equivalent fractions and how they can be used to find sums of unlike denominators.

This visual helps students see how 3/6 + 2/6 is another way to write 1/2 + 1/3.

Enrichment

For those students producing work samples like 5 and 6, we offered them a deeper problem to tackle in the remaining time:  Find the sum of 1/4 and 1/3.  We saw some interesting stuff.  Notice how this student’s way of drawing the fractions flummoxes him:

Some students reasoned this way.  Notice his choice of a common denominator.

We also saw students grinding out a magical algorithm again.  I saw the student work (in pencil) and asked him to tell me his thinking.  I scribed it below in pen.

I still wonder how fluent his understanding really is.  He was so solid on the procedure and had some sense of “how the numbers should work” and when to multiply and when to add, but he couldn’t show me with pictures why the denominator was 12.  And this matters.  It’s in the standards that students use equivalent fractions to add and subtract fractions with unlike denominators, not just the algorithm.

Beyond

After sharing the “Work Sample 6” showing how to split the fractions into sixths, we invited students do to it themselves.  They practiced drawing horizontal lines and justifying to themselves why the answer might be 5/6.

To bring closure to the lesson, we returned students to the objectives about persevering through a problem and communicating their thinking.  We selected some (not all!) of the student work above to scaffold a 5 minute conversation about the learning that had happened in the room.  We started with visual proof that the sum was not 1.  We then shared visual proof that it appeared to be a fraction just less than 1.  We unpacked why 3/4 did not seem like a likely candidate.  And then we invited visual explanations for why the answer might be 5/6.  Then we played the final video to reveal the answer.

 

Students applaud!

Closure

We brought the class to a close by asking students about what they learned today and taking volunteer responses.  We then offered them an exit ticket to collect some data about their ability to add with like denominators (visually), interpret a visual using a number sentence with fractions, and think of a word problem that contextualizes the number sentence.

We noticed that most students did fine on the first two tasks, but that students struggled to come up with their own word problems and contexts.

Reflections, Takeaways, Analysis:

The teachers chose the following research question as a way to focus their thinking:  How do we reach the needs of struggling students while also pushing forward and keeping stronger students challenged?  We wanted to design a lesson to help make math more accessible and meaningful for students at all levels.  For the most part, this lesson met that goal.  In both classes, engagement was always at about 95-100% for the watermelon number talk.  For the Black Box 2, the participation was about the same for about 10 minutes, and then some students started to tire of the question and get distracted.  But the open questions, the low floor, the use of visual prompts were all essential practices that maximized student engagement.

As one teacher put it:  If we want students to drive the discourse, they need a task that is engaging, accessible, and attainable.

One last important things the teachers noticed.  This lesson worked so well because of the image we chose for the number talk.  This image launched the students in the right direction and framed the learning experience perfectly for the Black Box activity.

(Note:  The teachers also had time to look at other great visuals and activities for 5th grade fraction and decimal operations.  We created this document that you’re welcome to check out.)

An Invitation

Give this lesson a try.  What did you learn?  How could we have made it better?  Let’s get better together.

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