Array-Bow of Skittles and Multiplication Strategies

It’s my hope that there are other elementary teachers out there that might find this analysis useful if they want to use this compelling and fun lesson by Graham Fletcher in their classrooms to engage their students in exploring multiplication reasoning and problem solving strategies (4.NBT.5).  It’s appropriate to use before and/or after students have explored two-digit by two-digit multiplication because it offers several pathways to a solution.  Furthermore, it can create the intellectual need to develop more multiplication skills for students moving forward.

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!  Pun totally intended.)

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 conducted a lesson inquiry using Array-Bow with two teams of 4th grade teachers.  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 50-60 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.

This lesson was taught to five classrooms of 4th grade students.  One class had not yet explored two-digit by two-digit multiplication.  Two classes had spent about a week and a half in the “multiplication chapter” of their curriculum.  The other two classes were taught a month later in the year and had already studied (and been tested on) two-digit by two-digit multiplication.  Because students were in such different places, the work student work samples you may see, select, and sequence for discussion may not contain all that is shown here.

One critical thing we noticed, the students that had been learning about different multiplication strategies (distributive property, area models, partial products) still struggled to use these skills to investigate and solve this problem.  Many students set up 58 x 14 formally, but if they didn’t know how to do that algorithm, they were stuck.  Most students who set up the algorithm but couldn’t solve it had to be prompted to think about other ways to represent the multiplication.

It was a wake-up call to us that students need to constantly be learning tools and strategies in problems solving contexts that are accessible to all learners.  It’s not enough just to teach the routine problems in the textbook.

Our objectives (as teachers):

• We want students hooked, engaged, motivated, positive, energetic….and to persevere in solving a problem.
• We are curious about:  In what ways do students know multiplication?  In what ways do they approach problems?  How well does student estimation promote interest in computation?
• We want to collect data about how well this 3-Act lesson format engaged/inspired ALL students regardless of skill level.
• 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.

Learning objectives (for students):

• You will be engaged and collaborate with each other to solve a problem.
• You will use a variety of strategies to solve this problem.

Some quick notes here…  We wanted to keep the language very simple and focus on their thinking and collaborating skills.  While we were looking for evidence of multiplication strategies, we did not want to burden the front of the lesson with too much vocabulary and we also wanted to keep the lesson open for multiple pathways.

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

Into (Act 1):

Hello everyone.  Today, we’re going to look at a short video and we’re really curious what you notice and wonder about it.  We’re interested in how you talk about your (mathematical) thinking with each other.

We played the short Act 1 video showing the jar fill up with Skittles.  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.

To help discussion, we put sentence frames on the board:  “I notice…”  “I wonder…”

All of our classes wanted to see the video several times.

What did you notice?  

We used the following prompts in reply to students with useful “noticings” such as “I see a jar filling with Skittles” and “He uses several bags.”

Interesting.  Can you tell me more about that?  Who else noticed the same thing?  Can anyone say it a different way?

This phrasing encourages students to express their thinking.  The questioning keeps more students engaged and helps to promote more student-to-student discussion.

I’m really curious.  What did you wonder?  What questions do you have?

Students ask a variety of questions.  We scribed all questions on the board.  We had decided ahead of time that we would sort the questions into two “lists” on the board.  We put questions about the quantity of the Skittles in one list and other questions in another list.  We obviously didn’t tell students this, but we wanted to be able to quickly parse out the key math questions (How many…bags, Skittles, Skittles in a bag) from all the others (Why did he do it?  Did he eat them all?).

When students asked a key math question, we followed it up with:

Interesting.  Can you tell me more about that?  Who else wondered the same thing?  Can anyone ask the question in another way?

Once the question “How many Skittles are in the jar?” came out and everyone understood it, we didn’t spend too much longer on the Wonder phase.  Sometimes the question “How many bags of Skittles did he put in there?” came out as well.  This is a useful question in a moment.

Hmmmm.  I curious.  How many do you think are in the jar?  Can you make an estimate?  

We gave them a moment to discuss in groups, make an estimate, and write it somewhere on their sheet.  Teacher walked around and scanned student work for estimates to get a feel for their values.  In our experiences, almost every class considerably underestimated the number of Skittles.

How many think there are less than 200 Skittles in the jar?  (You may find that a different number may be more useful to start this conversation based on what your students come up with for estimates.)

That means the rest of you think that there are 200 or more Skittles in the jar?  Make sure all others raise their hands.

Keep your hand up if you think there are more than 300?  Notice if hands go down.  More than 400?  More than 500?  We usually didn’t have many estimates greater than 500, but your experience may vary.  We stopped once there were three or four hands still up and then asked them for their estimates and wrote them on the board.

Wow.  That’s a wide range of estimates.  We have estimates less than 200 and all the way up to ____.  If you wanted to make your estimate about the number of Skittles more accurate, what information would you like to know?

Students pair-share.  With some minimal discussion, students realize that they need to know how many bags of Skittles are in the jar and how many Skittles are in each bag.  We scribed both questions on the board below our key question about how many Skittles are in the jar.

Why might that information be helpful in answering our key question?

This was sometimes a useful follow-up question to help us understand and assess student thinking.

At this point, we made a decision about how to reveal the rest of the information and begin Act 2.  In some classes, students had burned a lot of mental fuel to get to this point.  In those classes, we just showed the images that contained the information (14 Skittles in a bag, 58 bags total).  In some classes, we got to this moment quickly (within 10 minutes), and we decided to ask students to make estimates to these questions first.

I think you have enough information to answer our key question.  Let’s dive in a do some math and see if we can find a more accurate estimate for how many Skittles are in the jar.  

Through (Act 2):

What happens next will depend heavily on the skills and mindsets of your students.  I tried to capture a lot of the student work we saw in all five classes; I missed quite a bit, especially in those where I was instructing.  You probably won’t have time to share all of these work samples in a discussion at the end of Act 2.  You’ll need to make your choices about what work to share based on the needs of your students.  Let us know what you do and how it goes!

We let students dive in for 5 minutes.  During this time we monitored the room and gathered as much data as we could.  Did they understand the question?  What are they writing?  What are they saying to each other?  How are they making meaning?

In some classes, we needed to stop the student work time after about 5 minutes because their struggle wasn’t productive.  During this stoppage, we shared a few samples of student work (usually a repeated addition sample and another sample showing the multiplication algorithm).  We had students share their reasoning behind the repeated addition.  For the multiplication algorithm, we shared a way to use an area model as a tool to organize some thinking.

In some classes, there was enough collective energy and collaboration that students didn’t need this group check in.  For students who arrived quickly at 812, we asked them to show their reasoning in another way or look for another approach to solve the problem.  We encouraged groups to compare answers with groups and discuss their thinking.

For students who were struggling, we explored offering some or all of the following options.

Option 1:  What if there were 60 bags in the jar?  Show me how you could figure that out.  Then when they arrived at 840, we asked them what they could do next since they had added two extra bags.

Option 2:  What if there were only 7 Skittles in a bag?  How many Skittles would that be in total?  When they arrived at 406, we asked them how they could use that answer to answer our key question.

Option 3:  Show the 14 Skittles broken down in to a group of 10 and a group of 4.  Then ask them to figure out how many Skittles would be in 58 groups of ten and in 58 groups of 4.  How could we use those facts to figure out our answer?

We often saw struggling students choose the brute force approach of adding up 14, 58 times.

We interrupted their work before long, unpacked their thinking, and applauded their efforts.  We helped them to see that this was an effective strategy, but involved a lot of tedious work.  (They readily agreed!)  Could there be a more efficient way to do repeat addition?  Most realized they could add 58, 14 times.  We sometimes had students using the repeated addition strategy to pair up in class, share their thinking, and check to see if they calculated the same answer.

This student was showing the repeated addition model and also the distributive model they had just learned the day before.  It’s also interesting to note that 14 groups of 58 yields the correct answer, but does not most accurately reflect the scenario.  Regardless, it’s an efficient use of mathematical structure.

A lot of students would set up a repeat addition problem and then realize that it wasn’t going to be a lot of adding.  So they tried the multiplication algorithm, but didn’t have the conceptual understanding to carry them through the steps.  We noted (somewhat distressingly) that too many of these students couldn’t use other strategies to find the product of 58 x 14 when they couldn’t do the algorithm.

We saw the distributive property used poorly.

And also saw it used well.

One thing we noticed is that they almost all chose to break 58 apart instead of 14 when they used the distributive property.

There were a few students who were able to set up and complete the algorithm, but we were wary about their level of conceptual understanding behind the procedural skill.

In your classes, you will probably see more students using area models and these should be shared before the standard algorithm as they show the partial products visually.  The lack of many visual models in our classes was strongly noted.  One key takeaway all teachers had was the need to continue to scaffold learning experiences with more manipulatives, drawings, and area models.

Once productive struggle had been maximized, we started to share student work samples in the order sequenced above.  Whenever possible, we tried to share some visual examples.  We encouraged students in the class to connect to the presented work by asking:  Who did a similar approach to this student?  What more could you add?

In one class, we concluded without showing the big reveal (Act 3) because students were struggling so much.  The teacher concluded the lesson sharing some student work and ended with a reflection (and on a positive note!).  She wanted to continue the lesson again after some more work with multiplication.  Her students were eager to tackle Array-Bow again.

Beyond (Act 3):

So, it appears that the expression 58 x 14 models this problem.  And that the product of 58 x 14 is 812.  Let’s pause for a moment.  Why might 812 not be accurate?  What assumptions are we making?

This question prepares students for the fact that the answer is not 812.  It also creates space for a discussion that our goal was to use math to make a more accurate estimate, and we’ve done just that!  The math answer of 812 is a lot closer than their original estimates.  This is a key point I stress with students because it’s about the utility and limitations of using math to make models.  Math is something we can use to solve some problems, not just something that happens to us as students.

After some table and whole class discussion, most students realized that there might not be 14 Skittles in every bag.  Interestingly, we noticed that it was often the struggling math students that realized this fact more frequently in the discussion.

We played Graham’s Act 3 video.  (Love his new video!  Thanks Graham!)

We paused the video to freeze the final image.  Students spent some time working out the answer.

At the beginning of this lesson, you made an estimate.  Let’s take a look at that list.  Who had the closest estimate?  Then we used some more information and some math tools to come up with 812 as an estimate.  That’s a lot more accurate to the real answer of 853.  

In fact, the math calculations are almost always more accurate than every original student estimate.

To bring closure to the lesson, we reflected with the students.  We asked them to choose one of the questions below and write about it for 5 minutes.

• When did you feel successful during this lesson?
• What good ideas did I have today?
• In what situations could I use the knowledge that I learned today?
• What “light bulb” moment did you have today?

Volunteers then shared their thinking.

Reflections, Takeaways, Analysis:

• All of our students have different needs.  Tasks and investigations that are open like this are great for framing future content.  As a result of doing Array-Bow first, the problem can be revisited each time students learn a new way to reason about multiplication.
• This task creates the need to learn more about multiplication.  Repeated addition is such a headache!  Multiplication strategies are the aspirin!
• This lesson is accessible for students regardless of English fluency.  The Act 1 video is so compelling that the problem presents itself in any language.  Teachers noted that most English Language Learners were more engaged and talking more in this lesson than in any other previous math lesson that year.
• 3-Act Math lessons like this one create and foster more student curiosity and inspiration, especially for struggling math students.
• Teachers need to focus more on the progression of multiplication in their instruction and lesson plan design.  (Want to see Graham’s cool video about the progression of multiplication?  Of course you do.)
• We believe that we learn far more from our mistakes than our successes.  As a result, we chose to share a lot of student work that had errors.  We made sure that we celebrated the work and effort of each student that shared an error.
• Celebrating mistakes is a way to give a different sort of feedback to students.  It can shift the classroom culture to more of a growth mindset.
• If it matters to us as teachers that students can tell us what they’re learning, we need to make sure that lessons are structured so that they can.

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