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 division reasoning and problem solving strategies (3.OA.2, 3.OA.3 and 3.OA.7). It’s appropriate to use before and/or after students have explored division and allows for many different conceptual approaches to a solution including using repeated subtraction/addition, equal groups with or without manipulatives, number lines, arrays, bar models, and multiplication or division equations to model a real world problem. Furthermore, it can create the intellectual need to develop more formal (and efficient) division 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 amazing 3-Act Lesson!)
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 Knotty Rope with a teams of 3rd 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 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 3 classrooms of 3rd grade students. The classes had completed one (of two) chapters on division from their (not terribly engaging or fun) big-publisher curriculum. This chapter had explored the conceptual representations of division including repeated addition/subtraction, arrays, equal grouping, bar models, number lines, and formal equations. We used this lesson as a way to formatively assess student conceptual understanding of division and also their ability to persevere, communicate reasoning, and use appropriate tools strategically.
This lesson 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 all students hooked, engaged, motivated, positive, energetic….and to persevere in solving a problem.
- We are curious about: In what ways do students know division? In what ways do they approach problems?
- 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 and communicate a variety of strategies to solve this problem.
Some quick notes here… We wanted to keep the language very simple and focus on student thinking and collaborating skills. While we were looking for evidence of division 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 Graham tying three knots in the rope. 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.
What did you notice?
We kept this short. Once students all agreed that “somebody was tying knots in a rope,” we moved forward on.
I’m really curious. What did you wonder? What questions do you have?
Students struggled to come up with the key question that would move the conversation forward. If we were to do this lesson again, we would ask:
So you see a guy tying some knots in a rope. What do you think will happen next?
We would expect students to suggest that he will probably keep tying more knots. And then we could follow up their conclusion by posing the question ourselves:
How many knots will he be able to tie in the rope?
We scribed this key question on the board and asked students to watch the Act 1 video again and make a prediction (estimate) about how many knots he could tie.
We gave them a moment to discuss in groups, make an estimate, and write it on their graphic organizer. Teacher walked around and scanned student work for estimates to get a feel for their values. Then we asked them to share their estimates and we recorded a range of them on the board (usually about 6 to 50 in most classes).
Wow. That’s a wide range of estimates. We have estimates less than 10 and all the way up to 50. If you wanted to make your estimate about the number of knots more accurate, what information would you like to know?
Students struggle with this as well at the beginning. We’re striving for them to realize that they need two pieces of information: How long is the rope? And how much rope does it take to tie a knot?
We let students pair-share and discuss before attempting a group discussion. Most students see that they need to know how long the rope is (44 inches).
When they struggled to come up with the second question, we asked:
Do you think he could tie 100 knots? NO! Why knot? Because there isn’t enough rope!
We had students discuss and unpack that response a little bit and most students realized that some amount of rope was being used up for each knot that was tied (4 inches).
We displayed the two images that answer these questions. We allowed them some discussion time because it’s not immediately obviously what the image with the 40-inch piece of rope is communicating. If we were to do this lesson again, we would bring in two pieces of rope that are the same length. We would show their equal lengths and then tie a knot in the end of one of them to show how the rope shortens and that the difference is the amount of rope it takes to make one knot.
Through (Act 2):
This problem sets up an open opportunity to conceptually explore how to solve 44 ÷ 4. We let students dive in for about 10-15 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?
We provided some more intervention for really struggling students. The teachers had posted the different strategies they had used for division, and we often asked students to reference that for possible starting points.
For students who arrived quickly at 11, 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.
Here’s a quick snapshot of the work we saw (and work you should anticipate seeing).
We saw some adding up.
We saw some skip counting (blended with a bar/number line model).
We saw some repeated subtraction. I saw this student using her fingers to count down by fours. After each 4-count, she made a tick mark. She eventually counted out 11 groups. I asked her to explain her process and ask her what she was doing. The subtraction work you see here is me scribing her words. It was interesting to hear her count down using her fingers until she got down to about 20 and then her fluency took over and she stopped using fingers and could skip count.
We saw students making equal groups with manipulatives.
We saw students drawing groups. (Notice the difference in grouping compared to the example above.)
We also saw that some students showed conceptual understanding, but needed to pay more attention to precision.
We saw students use a more traditional bar method.
This student had some interesting approaches. She made her diagram using 11 fours, but then she counted the knots in her diagram and came up with 12 for an answer. We had her compare her work with other students as a method to figure out her error. She also used some interesting addition of chunks approach that you can see.
We saw a few arrays like this one.
We saw students using some related math facts to make sense of the problem.
And then we had some stronger students eagerly take on the challenge to show their reasoning in as many ways as possible.
Beyond (Act 3):
Once productive struggle had been maximized, we started to share student work samples. Whenever possible, we tried to share examples of actual student work. 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?
One of the things that makes this lesson so awesome are all the pathways to solutions. It’s also highly unlikely you’ll have the time (and the students will have the mental fuel) to go through each solution method.
We concluded by asking students to think about the method that they liked most. And then had them reflect on a method that they liked but wanted to practice more so they could get better at it. These types of reflection questions are helpful for students to connect the work of others to their own thinking.
We played Graham’s Act 3 video. For the most part, students were leaning in and eager to know what’s going to happen next.
Reflections, Takeaways, Analysis:
- All of our students have different needs. Open-middled tasks and investigations like this are great for framing review and sharing strategies.
- Knotty Rope allowed for us to see what all students could do. It allowed students of all levels to engage in thinking and discussion.
- The expectation to “find your solution using as many strategies as possible” differentiates this lesson and allows stronger students to reinforce their fluency and perseverance.
- This task will feed very nicely into Graham’s Orange lesson. The Orange lesson creates the need to learn more about division and find more efficient methods. Repeated addition and subtraction are such headaches when you need to find 165 ÷ 3!!
- 3-Act Math lessons like this one create and foster more student curiosity and inspiration, especially for struggling math students.
- Teachers need to focus on the progression of division in their instruction and lesson plan design. (Want to see Graham’s cool video about the progression of division? Of course you do.)
- 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?