This is a post in a series about problem based instruction in fourth grade — centering on a series of lessons classroom teacher Claire* and I taught about symmetry. This is part 4.

At this point, Claire and I had taught two lessons on symmetry in Claire’s fourth grade class. The second lesson focused on using what students knew about 2D geometric figures to make conjectures about symmetry. The third lesson in the sequence would allow students to try out one another’s ideas in a new context: quadrilaterals.

- Lines of Symmetry for Quadrilaterals (task from Illustrative Mathematics)

As Claire and I planned, we kept in mind the “five equity-based mathematics teaching practices” from* The Impact of Identity in K-8 Mathematics: Rethinking Equity-Based Practices* by Julia Aguirre, Karen Mayfield-Ingram, Danny Martin (NCTM 2013)

- Going deep with mathematics
- Leveraging multiple mathematical competencies
- Affirming mathematics learners’ identities
**Challenging spaces of marginality**- Drawing on multiple resources of knowledge

### Selecting students to share about the quadrilateral task

Thinking from three different students had been shared the previous day. In selecting students, we were mindful about each student’s context. Giselle is seen as high status in class, and has a lot of prior experience about the content from formal math classes she takes outside of school. Sara is someone who will often share her thinking, but then second guesses it when others question it. Avi is a native speaker of Hebrew; he moved from Israel in second, and participated in our English Language Learner program.

Claire and I thought hard about whose voices we wanted to amplify in the whole class discussion about the next task. Students will have already seen some strategies using side length and angle measure that could work again, so this task felt like it was more consolidation and application than trying out a new concept.

From “Lines of Symmetry for Quadrilaterals,” *Illustrative Mathematics*

### The Quadrilateral Task

For our warm up, we used a true or false set of equations. We wanted to activate thinking about equality — being equal on both sides, like the line lengths and angle measures.

After launching the quadrilateral task, Claire and I observed students at work, and conferred with them individually and in pairs. We did not see any students cutting and folding today. Most seemed to be focused on side length.

Because so many students were using the same strategy, Claire wisely decided to keep this group discussion brief. (Sometimes, I stretch it out, and I need people like Claire to keep me in check!)

### Amplifying Student Voices

Claire did make note of several students whose voices she wanted to amplify. This included Ezra, who I wrote about in lesson 1, as well as Harry, a new student who had moved recently from China. Harry smiled a lot, but didn’t talk much in class. It didn’t help that his body was so much larger, in both height and weight. He stuck out for so many reasons. The class seemed to embrace him immediately, but Claire worried that they weren’t taking the time to get to know Harry. He felt more like a class mascot.

Harry also had some great thinking. Claire decided that she would draw the lines of symmetry, and that Harry would try to use words to explain them. “If he just says up and down, that’s fine,” Claire told me. “I can work with that!”

He did. Claire drew the lines, and then added in some more precise language. “Oh, so right through the middle, from the center of the top line to the center of the bottom line.” Claire then helped the class check using side lengths, and angle classification.

The conversation advanced quickly, with Ezra and Harry in the lead. We then had time to launch a second task.

*originally published on the Illustrative Mathematics blog – *“Designing Coherent Learning Experiences K-12,” *by Kristin Gray, May 8, 2019.*

### The Circle Task

This is what Claire viewed as the pinnacle of the symmetry content. This task gets at the idea of having an infinite number of lines of symmetry. It’s why Claire thought her previous students had missed a question about symmetry on our state test. “I could see that the greatest number of lines of symmetry is a circle, so the shape that’s the closest to the circle would be the one with the greatest number of lines of symmetry. But my students were used to cutting out shapes and folding them. They couldn’t cut out the items on the state test.”

The circle task allowed students to wrestle with this. It also asked students to engage in standard for math practice #3 (SMP3): construct viable arguments and critique the reasoning of others. Students would need to

from Lines of Symmetry for Circles, from Illustrative Mathematics

In order to be successful with this task, students will need to construct their own arguments, wrestle with Lisa and Brad’s ideas, and also use ideas from Lisa and Brad to support their own conjectures.

### Students at Work

Claire and I were excited to see students writing during this lesson. The previous tasks had involved high student engagement, but not a lot of writing to express and consolidate their ideas. Students would have to write here.

Yu had come to our school midyear from China. She expressed herself well in English, but she wasn’t fluent. I loved that her work was often peppered with Mandarin words like this. I took this photo while she worked, and Yu helped me translate the words using a translator app on my phone.

How many lines of symmetry does a circle have?无穷 (“Wúqióng”)

Infinite.

The other character used, 折 (“Zhé”) means *fold*.

More students were starting to think about the angles and sides of a circle, a shape that Claire’s class had not previously explored in any depth.

*Yes because a circle has no angles so any line that goes through the full shape is symmetrical.*

*Yes, because if you folded it down the middle, both sides would match.*

How many lines of symmetry does a circle have? *Infinite lines of symmetry, because a circle has no sides or points.*

### Whose work/thinking anchors the discussion?

Claire chose a few students to share at our group discussion, or *activity synthesis*. (We used the five practices for orchestrating classroom discussion to guide selection.) Claire wanted to pull together ideas about the infinite nature of the circle. Does the circle have no points? Infinite points? No sides? Infinite sides?

While keenly attending to the mathematical ideas, Claire also thought about which students are not as vocal or represented in math class. She wanted to give them a chance to ground our discussion.

Hannah shared first.

I wrote about Hannah in the post on the first lesson in this sequence. She is legally blind and uses an ONYX, a portable video magnifier, to help her access some visual components of the classroom environment. The ONYX machine allows her to zoom in on the board, change the contrast, change the coloring, etc. In previous years, I had seen Hannah disengage from math class. She would squint at the page, or try to trace it. Now, with enlarged copies of the tasks and ways to interact with it, Hannah recognizes herself as a member of our math community.

Hannah has always had ideas worth sharing, but now she is able to develop them — and share them!

While Claire had chosen Hannah deliberately, and hoped to elevate her status within the math classroom, she did not make a big deal about Hannah sharing. This felt like business as usual.

[the example is correct]because it goes directly through the midle and seperates it into 2 semie circles.

Hannah attended to the fact that both of the fictional student work samples passed through the center of the circle, an important feature of the lines of symmetry. While there are infinite lines of symmetry in a circle, not every line drawn through a circle is a line of symmetry.

Then came Jackson’s work.

[A circle has infinite lines of symmetry] because a circle has no angles it also means there’s infinite angles so if there’s infinite angles theres infinite lines of symmetry because you can keep cutting the circle in half infinite times.

[each line of symmetry divides the circle in half] because it divides in half to prove the thinking you can fold a perfect circle. You’ll find you can fold it infinite times in half.

Jackson helped make the connection back to the folding work students did in lessons 1 and 2. He also started us thinking about the properties of the circle.

“Does it have infinite angles or no angles?” Arjun asked.

“How can it have an angle if it doesn’t have a side?” Avi asked.

Great questions! Claire and I wanted the students to try to answer this, although the conversation kind of danced around it. So Claire put Sophie’s work on the projector.

Sophie is friends with Giselle, and seems to defer to her mathematical knowledge when they partner. Centering her as an important piece of this conversation has the potential to build her mathematical identity.

[each line of symmetry divides the circle in half]because a circle either has an infinite number of sides or no sides if a circle has an infinite number of sides then you just keep dividing in 1/2

Sophie’s work did satisfy Arjun and Avi with an answer. A few other students seemed to be squint and nod, perhaps trying to process this debate between no sides and infinite sides.

Meanwhile, Ezra turned away from the board, completely disengaging from the discussion.

Claire had selected Liam’s work for last. Liam didn’t write much, but he had made a diagram. Often a diagram can help clarify ideas. We often put them first, moving along a progression of “concrete/representational to abstract,” although there is often power in putting them last.

Arjun cocked his head to the side. “I think that this proves there are no sides. You can draw the lines anywhere.

“Or maybe it show how there are millions of tiny sides that you can cut through.”

“I was thinking no sides,” Liam clarified.

“But it doesn’t have to be no sides,” Giselle countered.

Most students had written some ideas about either infinite or no sides/angles in their written work before this discussion. This students who spoke the most during the actual discussion often had higher status. The students whose work anchored us were not always among their elite, more vocal crew. It made for a good first start into this problem-based space.

Claire and I could have incorporated some different facilitator moves — more turn and talks may have allowed other students an opportunity to rehearse their rough draft ideas, and feel empowered to share them — but we agreed that the class headed in the right direction.

“I don’t think this is a class that would get tripped up by that lines of symmetry question from the state test. We’ve gone so much deeper,” Claire reflected back later.

### Lesson Close

In the moment, Claire could sense that students weren’t any closer to consensus or a conclusion, so she reenergized them with a sneak preview of tomorrow’s work: a project about mandalas.

Mandalas typically have spiritual meanings, and Claire wanted to honor this. The class had already been having conversations about surface vs. deep culture during their social studies block. (Social studies was often right before math in the morning, so I was able to bear witness to some of this beautiful work.)

So Claire launched with a video about mandalas. Despite the fact that students had now been on the rug for a challenging 15-20 minutes of discourse, they were fascinated. Claire closed the video with a return to the notice/wonder routine she had employed in the first lesson.

Students wrestled with whether the radial lines stretching across the mandala represent all of the lines of symmetry. The different concentric ‘layers’ of the circle presented challenges. If there was no design in the circle, there would be infinite lines of symmetry, so every addition meant a decrease in lines of symmetry. Different ‘layered’ rings may have fewer lines of symmetry when put together than when separate.

Students would be dissecting, analyzing, and then creating mandalas in the days to come.

*In my blog, I use pseudonyms for all students, and most adults. ‘Claire’ helped choose her pseudonym.

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