I had a chance to use Steve Wyborney’s tile images as a part of a 3rd grade lesson inquiry. The tile problems are an effective tool to engage students in discourse about their mathematical reasoning. Furthermore, it allows teachers to identify student misconceptions about partitioning and calculating area. These misconceptions often prevent students from understanding how to use the area model to reason about fractions (3.NF.1).

I’d like to share with you what we learned, and I invite you to continue the dialogue with me in the comments section, particularly regarding input about how to address the student misconceptions we faced. I hope that this write-up could be used as a guide for any 2nd-5th grade teacher that wants to work on number sense and fractions (conceptually and/or procedurally) with their students. Please try it out and share what you learn.

You can find Steve’s blog here and his @SteveWyborney Twitter feed here. He’s got several posts about the tile problems. As of the writing of this post, you can find three of them here, here, and here. The videos are 6(ish) minutes long and are essential. Watch them! All the instructional resources (PDFs, PowerPoints, etc) are on his site.

## Some quick background:

I conducted a lesson inquiry using the Tile Problems with a team 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 60-80 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 students were about to start the chapter that introduces fractions, and we assumed that they had some working conceptual understanding of “whole,” “half,” and “quarter,” but did not have any formal understanding of fractions as numbers. We used this lesson to see what reasoning skills they already possessed to solve area problems by finding the value of partitions. These skills are the building block to understanding the area model for fractions.

*In a sense, we were trying to create a headache for students (how do we describe numerical values that are only parts of wholes?) that would frame the next chapter. We wanted students to realize that fractions were a useful tool for talking about such values.*

All students had a packet with the tile patterns we were exploring. Each page in the packet had 4 copies of an image. We wanted to students to use as many strategies as they could to find the value (area) of the figures.

## Our objectives (as teachers):

- What do they know? What misconceptions do they have? What approaches do they take?
- Teacher will monitor student learning, look for anticipated student 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 show different ways to make sense of area by using number sense strategies.
- You will communicate your reasoning with each other.

## Into:

**Good morning everyone! Today’s objectives for math are:**

**Find the value of shapes.****We want to see you show your thinking in as many ways as you can.****We want to hear you talk about it with each other.**

Note: We wanted to keep the intro short and use very simple language. We felt that “find the value of shapes” was the lowest floor that we could start on that engaged all students.

(All slides can be found on Steve’s site. We hunted through his collections to find the 5-6 examples we thought provided the appropriate level of struggle for our students. You can find our slides here.)

We projected the image on a white board. We wrote a large “= 1” next to the white square in the top left corner. And then said:

**The value of this small box is one. What is the value of the whole shape altogether? If you find a value of the whole shape, we challenge you to try to show how to find the value in as many ways as you can.**

We chose this image and “small square equals 1” because we felt it was low-floor enough that all students could make an attempt. We didn’t want to start the lesson with direct instruction. Students worked for about 2-4 minutes and then we had them pair-share in their groups.

**Share your work with your partner. Did they get the same answer as you? Did they figure it out the same way?**

We scribed sentence frames on the board: My answer was _____. I found it by ________ (tell us what you did).

The teacher floated the room and monitored student thinking/discourse and then chose 2-3 approaches to share.

We saw about 25-50% of the students correctly use a decomposition strategy to calculate the value (area) of the figure as 9 (square units). Other students struggled, and we were able to expose some misconceptions.

Damian’s work below shows a misconception we saw in every class. Some students wanted to count the shapes.

After students had a few minutes to work, we copied Damien’s example counting the five shapes up on the board in every class.

**Let’s look at this example. What approach did this student take? What could we tell this student to help their thinking? **Student responses varied, but we were striving to help students see that bigger shapes were worth more because they contained more small squares.

We then shared an example like this.

**What strategy did this student use?** He cut all the shapes into small squares and then counted them. Qasem’s work above is interesting because he has correctly decomposed the shape, but his writing suggests that he’s counting up rather than adding up. In other words, he doesn’t write 1 in every box and then add. He’s using numbers to count the boxes. We did NOT always unpack this in every lesson choosing to move the conversation forward instead.

This student’s sample shows examples of what we were hoping to see.

We focused heavily on the examples like the one in the upper right that shows all the “1”s.

**How many of you used a similar strategy?** We wanted to make sure that all students could make sense of this fundamental partitioning (decomposing) strategy because it was the key through-line for the entire lesson.

We then had students share grouping strategies. The work sample above shows two different ways to calculate the area by grouping shapes.

**So there are some different methods here. Cool. Try to use these methods to find the value of this next shape.**

We then let students practice these strategies on the second image. The small square had a value of 1.

We surveyed the room and examined student work to see if they all had a strategy (decomposing or grouping) that they could use. Success went from about 25-50% up to about 80-90%. We briefly had students share solutions with each other and the whole class, but we were also cognizant of the time. We wanted students to have enough “brain fuel” for harder problems moving forward.

## Through:

**So this problem is a little more challenging and different, but I think you can give it a good go.**

We showed the slide below and wrote “4” on the blue square.

**This time, the big blue square has a value of 4. What’s the value of the shape altogether? Again, try to find your answer in as many ways as possible.**

We wanted to slowly ramp up the thinking one small step at a time. Would they find the value of a small square? Would they make other choices? Students worked on this problem for about 5-7 minutes.

**Share your work with your partner. Did they get the same answer as you? Did they figure it out the same way?**

Teacher again floated room, monitored student thinking, and then chose 2-3 approaches to share. We wanted to link back to previous methods that were used but also keep an eye out for new solutions.

The work below shows some decomposition and grouping strategies.

And then there are some students that showed some creative partitioning.

In general, all students were able to use at least one strategy to figure out the area. We highlighted student language that was useful for later work such as “whole,” “half,” and “quarter.” We did not see students “making 4s” by grouping the smaller shapes into “big squares.” To address this gap, we showed the strategy on the front board and asked students to examine it and make sense of it in pairs.

Like before, we offered another figure that allowed them to practice these strategies. And it was this image that exposed a lot of misconceptions. Take a peek.

Here’s the next slide.

We said the big blue square had a value of 8. This is the first time that the small square did not have a value of 1, and students really tripped on what to do. We saw a lot work that showed some misconceptions. (I’m really curious about your take on these misconceptions. What misconceptions do you see? What would you say to these students?)

Woowee! As you can see from the work above, students really struggled when we changed the value of the small square to 2. How would you have sorted this work and used it instructionally? What choices would you make? This was a thorny stretch in all three classes for us as instructors, and we were a little surprised at how many (about 50-75%) of students struggled with this prompt.

One thing we noticed is that many students intuitively knew that they needed to cut the big blue square into eight equal parts (eighths). However almost all of those students who did that did NOT realize that they were finding the size of 1 unit and then use that size to find the values of the yellow and green shapes.

The example on the left below is a good example of where she cuts the big square into eighths and labels each part 1. But she doesn’t use that size to partition the other shapes. She’s still stuck that the small square equals 1. Was this because we had scaffolded using the “small square equals 1” too long? Was this a deeper misconception? If so, how would it affect her ability to work with fractions? What corrections should we consider as her instructor?

There were also promising samples of student work like the example below.

**What was funky/weird/different about this last example? **The small square didn’t have a value of 1; it had a value of 2. *And that affects the value of all the other shapes as well.*

With this tension in mind, we wanted to ease gently into our last and final problem. We stressed that we were proud of how they were showing and sharing their work and also talking to each other and persevering through some hard problems.

**Here’s another challenging one for you. We are really curious how you make sense of this problem and how you tackle it.**

**This time the blue square has a value of 1.**

By this point, most classes had used the words “half,” “whole,” or “quarter” to talk about the other images. We were hoping to see how students use those words to describe the sizes of the smaller shapes. We didn’t expect students to use fraction numbers (1/2, 1/4, etc), but we were hoping they used some language around describing parts that had a value less than 1.

*In a sense, we were trying to create a headache for students (how to do we describe numerical values that are only parts of wholes?) that would frame the next chapter. We wanted students to realize that fractions were a useful tool for talking about such values.*

Most students really struggled to reach a solution here. We weren’t surprised by that result. Here are some of their samples. What do you notice? Wonder? What would you say to these students?

We were also blown away by the skills that some students showed. I remind you that the following work is from *3rd graders*.

**Beyond:**

We concluded the lesson by having students share their reasoning about the problem above. We didn’t worry too much about refining thinking; we were way more interested in hearing what they were thinking about. We also repeatedly stressed to them that it was ok if they didn’t know anything about fractions and that they would have plenty more time to practice and play with tile problems like this.

We handed out an exit ticket that asked students:

What is something that you learned today about finding the value of shapes? Draw a picture to show your thinking. Here’s an exemplar response we were pleased with.

## Reflections, Takeaways, Analysis:

After each lesson, we reflected on our observations about student learning. And at the end of the day, we had a longer reflection on all three lessons. Here are some things we talked about.

- The tiles provide an engaging entry point for students at all levels.
- Students seemed to be pretty excited to find the value of shapes in different ways.
- The structure of the lesson allowed us to see what (almost) every kids was thinking. We could identify misconceptions and mistakes quickly.
- Students really enjoyed the opportunity to share strategies with one another. We heard a lot of students say “Oh! I get it!” to each other.
- Students also enjoyed when we used their work to guide conversation.
- We talked a lot about the scaffolding in the lesson and how it builds. We liked having two low-floor problems at the beginning. The first problem was challenging for many but they were able to pick up a strategy quickly and move forward.
- Obviously, the big hurdle for them came when the small square no longer had a value of 1. This choice forced us to think critically about our instruction and our questions to students. (Please chime in the comments!)
- Teachers felt like they could use problems like these regularly in the weeks to come. And that they were easy to create and easy to differentiate for students.
- Another teacher said: “I see the value in presenting lessons where kids are exploring their knowledge and then working through the problem; lessons that lets students wonder ‘what am I thinking’ rather than ‘what am I being told’.”
- Another teacher: “It’s possible to scaffold a learning experience without much direct instruction. What’s the appropriate level of instructional guidance? What balance maximizes learning for all? For individuals?”
*Most importantly, we never saw a time when students demonstrated a fear of being wrong.*

And that last point is probably the most valuable thing about using these tiles: kids take risks, do math, and then talk about it!

## What we might do differently next time:

- Pay careful attention to the scaffolding of the problems.
- Have a blank word bank up on the wall that we could add to as students used useful math words/language. The word bank could then be refined in future lessons.
- We saw a lot of creative partitioning of shapes. (I didn’t show that in the work above much.) We wondered how we could use this creative talent in future lessons.

We’d love to hear your thoughts and insights. Let’s keep the conversation going and get better together.

This may be the best representation/overview/debrief of a lesson I have EVER seen! Thank you for such clear explanations, the script, student examples and exemplars, comprehension percentages (!)–I feel like I just observed this and debriefed it in person with colleagues. I can’t wait to use it!

I’m so glad you found it useful Nadine! Please let me know what happens and what you learn. Let’s get better together!

Thanks so much for the feedback.

Fascinating. Students were thinking and figuring out different values and not focused on “the right answer”. The work and descriptions reflect a culture and atmosphere of trust. Kudos to you and your students!!

Thank you for this very clear summary of how these lessons were conducted. I am VERY excited about taking it back to my team.

Thank you! Please let me know how it goes, what you learn, and how we can get better together.