I conducted a lesson study with some elementary teachers. We used Dan Meyer’s engaging lesson called Sugar Packets to get students talking about an interesting problem, sharing their thinking, showing what they know about division strategies. Dan has the lesson listed as 6th grade ratio and proportional reasoning activity, but we found this problem to be suitable for both 3rd and 4th graders and possibly as a review for 5th graders. There is a remainder in the solution. And we found that this lesson worked brilliantly for students who haven’t had many opportunities to learn about remainders. It’s a wonderful introduction to thinking about the contextual and mathematic meaning for the remainder. (If you teach 3rd grade, I think you’ll find that your students will dig it! Don’t let the remainder spook you off!)
This lesson addresses many of the Operation and Algebraic Thinking standards for 3rd and 4th grade. It is also a rich opportunity for students to reason abstractly and quantitatively and to communicate their reasoning with each other.
So, give it a read and give it a go! Let us know what you learn. Let’s get better together.
Some background:
I conducted this lesson study with a team of 4th grade teachers. It was taught to 3 classrooms, and each lesson was a unique experience with its own twists and turns and choices. What we show here is the result of how we revised the lesson throughout the day based on our observations of student learning. 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 division. The students hadn’t seen division since 3rd grade and we wanted to use this problem as a pre-assessment to see what skills students would show us. The data would help inform our instructional choices for the upcoming division module in the curriculum.
If you haven’t already, take a peek at the Sugar Packet video and resources here.
Our objectives (as teachers):
- We wanted all students to be engaged and demonstrate “math confidence” by asking and answering questions, demonstrating ownership and initiative, and helping other.
- We wanted to see students enthusiastically engaged in discovery, sharing their thinking, and celebrating “Got it!” moments.
- We were curious to see if this open lesson format helped to make math more accessible and meaningful for all students. We wanted a pre-assessment that allowed for students to show us what they can do rather than what they can’t do (which is what a lot of pre-assessments measure).
Learning objectives (for students):
- Today, you will work together and share your thinking to solve a problem.
Into
(Note: We did not focus heavily on the objectives at the start of the lesson, but just stated that we were interested in how they communicated their thinking and how they listened to each other. We used the Into portion of the lesson as an opportunity for students to internalize the objective about sharing their thinking about some basic multiplication and division problems.)
We started a number talk 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 two problems, spending about 5 minutes on each one talking about different strategies:
6 x 10 and 66 ÷ 11
We were hoping that students would link the second problem to the first. As students did this, students were able to make explicit connections between multiplication and division. The emerging strategies would also be useful in the upcoming task.
Through
Students moved back to their tables where they had access to their whiteboards and some base-ten blocks.
Take a look at the following video. What do you think this video is about? What questions do you wonder about?
(Click here if viewing on your phone.)
Students had an opportunity to pair-share and talk about what they noticed and shared what they thought the message of the commercial was about before we facilitated a quick whole class discussion. A common response in 4th grader speak: “It’s funny because he’s eating sugar packets and the other people are looking at him like he’s weird, but they don’t realize that they’re eating (drinking) a lot of sugar too!”
How much sugar is in soda do you think? How many sugar packets do you think are in a bottle? What would you estimate?
Estimates often ranged between 10 and a 100.
If you wanted to know how many sugar packets are in the Coke, what would be useful to know?
We gave students a time to share their thinking in groups first. This step can be a sticky one for some students, but it’s important to ride it out as a teacher. If we want students to make meaning of a problem, we need to withhold some information so that student thinking can fill in the gaps.
Students realize that they need to know how much sugar is in a sugar packet and how much sugar is in a Coke. We shared this information only after students asked for it. You can display the images on Dan’s page here, but we actually had a 20-oz bottle of Coke and some sugar packets and invited students to read the labels. (We checked to make sure the sugar information matches what’s in the problem, 65 grams for the Coke and 4 grams in a sugar packet. They did, but not all sugar packets are the same.)
Armed with this information, we turned the students loose and invited them to show their thinking on their whiteboards or using the base-ten blocks. In our lesson planning, we watched this amazing video about the progression of division by Graham Fletcher. Using this information, we had anticipated a wide range of student strategies including:
- adding up or skip counting,
- repeated subtraction,
- making equal groups using base-ten blocks, drawings, or place value charts,
- thinking about 4 x ?? = 65,
- using the division algorithm.
We monitored the room looking for student work of these examples. We were especially interested to see how they grappled with the remainder. We didn’t see it all, but here’s some of what we did see.
This student skip counted, but realized that skip counting landed on 64 and she needed one more. Skip counting was by far the most common approach by students. One student (work not shown) actually drew a quarter of a packet showing an accurate mathematical representation of the remainder in context. Awesome sauce!
This student (who was identified as a struggling student), patiently drew 65 dots (without much thought to order or structure) and diligently made groups of four. He then counted his groups. Note his interesting “17th group” and how it only contains 1 dot. We asked him why he did that and what it means. He said he needed a 17th packet but only a bit of it. Awesome sauce!
These students drew four groups and then started to count up to 65. They eventually arrived at the “remainder issue” and had a rich conversation about what that might mean. This strategy was also a common approach for students. Note the difference in structure between these groupings and the groupings above. The student above made a more contextually accurate picture (showing 16 groups of 4), but the students below (consciously or unconsciously) showed how to divide 64 into 4 groups and interpret the 16 dots of each group in context. It’s less contextually accurate, but useful math worth sharing and talking about.
This student found a pathway to a solution that we didn’t think of. He chunks his adding up, realizes he goes too high, and subtracts chunks down. He’s then able to translate between grams of sugar and number of packets (a foundational skill for proportional reasoning in 6th grade). This work is slightly incomplete and doesn’t show him adding on another packet to get to 64. We nudged him forward to completion after this pic was taken.
These two girls used manipulatives to make groups of 4. Note the yellow cube (the remainder). They had a productive discussion about what the yellow cube meant. This student work provided a powerful visual of the remainder and we used it to facilitate class conversation.
We also saw students using the formal algorithm, but I didn’t get a pic of that student work. We noted that we did not see many students ponder the related multiplication fact 4 x ?? = 64.
Beyond
When we felt like student effort and achievement had been maximized, we selected important work samples to share with the class. We sequenced those examples like such:
- Skip counting up to 64 and thinking about how to get to 65.
- Grouping using concrete manipulatives and talking about the connection between the “yellow brick” with the need for “one more” in the skip counting method.
- Students then looked at and compared different representations of the grouping (drawing groups) and we asked them to examine how the groups were different (16 groups of 4 and 4 groups of 16) and talked about the remainder as the “leftover dot” that doesn’t have a home.
- In some classes, we shared some work about the division algorithm.
All the while we are trying to connect their thinking with the content. How do these diagrams show division? What is similar about them? What is different? What does that left over part mean?
And finally: What appears to be the answer we are agreeing on?
Once students agree that the answer appears to be “more than 16 but less than 17 packets,” we shared the answer using the video below. Right before playing it, we focused them on the remainder by asking: Let’s see how they handle the left over part.
(Click here if viewing on your phone.)
We let students watch the end of the video again and made sure they noticed he had part of one more packet and connected that to the remainder.
Closure
We offered them an exit ticket asking them to use drawings or words to show why 35 ÷ 7 = 5 is true and asked them to come up with a word problem where that equation might be useful. We wanted to give them a calculated equation because we wanted to see what students could show us about their conceptual understanding of division.
Reflections, Takeaways, Analysis:
- This 3-Act math lesson design was highly effective at getting students thinking and doing math.
- Teachers recognized that practice makes for a smoother lesson that flows. They were eager to try more 3-Act Math lessons like Sugar Packets. (There are many online, but you can find several great ones on Graham Fletcher’s site.)
- The open middle of this lesson allows for more student interaction with the problem and with each other. When we create open lessons, we create more opportunities for student choice and voice to drive the classroom discourse.
- This type of task was good for front-loading an upcoming unit and seeing what students were able to do. This assessment allowed us to get meaningful data while also building and growing classroom culture. We got better data about how students think about division and the skills they had for solving division problems.
- We noticed that many students liked the grouping strategy and also skip counting. We wanted to start the division model with some problems where the dividend is too large for those methods because they are too inefficient. By putting students in new situations where previous conceptual understanding is exposed as being inefficient, we create the need for a more structural and abstract method (the division algorithm).
- We noticed that several students carried out a calculation and quit. Some students did not demonstrate math confidence and a willingness to persevere and make sense of the problem. We also noticed that a few students were still “adrift” in the problem and struggled to buy-in to doing it. That said, struggling students experienced more success than they would have if they were tackling a more traditional textbook problem.
- The remainder (the first time the students have seen one) was tricky but gave them a wonderful and interesting headache to talk about.
What did we miss? How could we make it better? We invite feedback in the comment section. Let’s get better together.
