If you haven't read Part 1, I highly recommend taking a look! In that post, I talk about how a commitment to building students' confidence in math might by the single most important thing we do as teachers of mathematics. You get to hear how one student transformed in Mrs. Jones' class. While my time with my colleague in third grade was filled with moments of joy, celebration, and fantastic reasoning about fractions, I'd like to highlight our first lesson of the week.

Considering the very few years, grades 3-5, where students are learning specifically about fractions and operations with fractions as listed in the standards, teachers must ensure students have fraction sense and can talk about fractions beyond identifying them in circle models. Below, I will highlight a lesson that comes from Julie McNamera’s and Meghan Shaughnessy’s book ** Beyond Pizzas and Pies** and how the manipulatives chosen for this lesson were a great tool for students to not only demonstrate their thinking, but also think about fractions with the manipulatives.

It had been a while since I had taught this group of students, so I asked the Mrs. Jones if I could start with a brief number talk to learn about student thinking and how they are currently making sense of fractions. Below is what was written on the board and students were asked: '*What part of the large square is shaded?*'

The class was divided into two camps, ⅓ and ¼. Some reasoned that since there were three sections and one was shaded, that it must be one-third. A small number of students agreed. However, there were a couple students adamant that there was no way it could represent a third because they learned that fractions must have equal size pieces. I needed to know more so I used the talk move '*say more'* to see if I could glean a little more information. Another student added that if you were to draw a line through the top, just as the bottom is, we could clearly see 4 equal size pieces of which one piece would be named one-fourth.

This did not fully convince Sam. He argued that '*you can’t just add a line*' and that '*we were only shown a shape partitioned into 3 sections so we call those spaces thirds*'. At this point, the class was at a stand still. One group couldn’t fully convince the other what the correct name for the shaded portion was, however I felt that I had learned a good deal about what students do know and what ideas we could continue exploring with the task they would do that day. I left the drawing on the board and our two answers for us to come back to at another time.

We were ready to dig into fractions. In this lesson that explores part to whole and whole to part, students would be using one of my favorite math manipulatives, the Cuisenaire rod. Since students had not had experience with these before, I asked them to start by spending time taking out various pieces and reporting back to the class some things that they noticed. I like to have students do this because children's natural curiosity and affinity to find patterns generally lead to great connections. When we reconvened, many students had already discovered many relationships between the different colors. A favorite among the group was to line them up in order from largest to smallest! I asked students to grab an orange rod and place it in front of them on their desk. They were then instructed to find the rod that is half of the orange rod. I did not expect this to be challenging for students and that proved to be true. Within a minute, all students had identified the yellow rod.

I then asked students to tell me how they know that the yellow rod is half of the orange rod. Students were asked to think about how they would explain that yellow is half of the orange and then turn and share with their neighbor. When we came back together, one student shared that since she could '*line the yellow up underneath the orange and it went to the middle that it had to be the one-half piece*'. Many students agreed with this statement. I pushed further and asked who could state this in another way. A different student added that since he could fit two of the yellow rods on the orange rod, the yellow would be half of the orange. By asking other students to use their own words to describe the same model, everyone in the room has the benefit of hearing this idea many times. I wrote the second student's response on the board as a statement and asked if students agreed.

Next, students were asked to find the rod that is one-fifth of the orange rod. I let students get to work right away knowing Mrs. Jones and I would need to check in with a few students about what it means to be a fifth. Students were able to articulate that it would be something that would fit across the orange rod 5 times. This again, proved to be fairly straightforward for students. (See the rods above) I allowed them to also find the rod that is one-tenth, as most students had already figured this out in their exploration of the Cuisenaire rods. Once we came back together, I wanted students to not only use language to describe what it means to be one-fifth and one-tenth, but also reiterate the idea that the pieces needed to be the same. For example, one student had lined up some random pieces that did in fact cover the orange, but they were not all the same color/length. I made sure to display this for students to talk about why it may or may not be considered a fifth. Students agreed that to name the fraction, the pieces needed to be the same size. (This felt like an appropriate generalization for third graders.)

Students were then asked to take out their brown rod. This time, they were to look for a rod that is half of the brown rod. While some students took the approach of digging out the rods and placing them underneath to check, some students had identified the rods by number. For example, the orange is 10 white rods long, and since white is the smallest, they called the orange a 10. Brown had been identified as 8 units long so students knew they needed to look for the rod that was 4 units long to find the half. Students recorded their findings and had language to describe how they know that the purple rod is half the brown rod.

Once students found one-fourth the brown rod and then one-eighth, we came together on the carpet to talk about our findings. I started by recording what we know in two tables below.

I asked students to think about why the yellow and the purple are both named one-half. As students learn about fractions, they often get stuck in one model or one tool that represents fractional pieces. It’s important that they have multiple meaningful experiences and especially with different representations and tools that help to solidify the big ideas around fractions. Mrs. Jones and I agreed that students need to recognize that the numerator and denominator are a relationship. The whole matters, and the numerator and denominator describe a relationship of the given whole. This holds true when we talk about parts of a set, or even ratios.

As students were pondering the question, Mrs. Jones drew two circles, two rectangles, and two number lines. Each one of the set was much larger than the other.

She asked her students to think back to when they were discussing fractions with food. If there are two pizzas here, one large and one small, how can we represent half of each pizza? Students were able to articulate that she should draw a line down the middle to show two separate equal size pieces. She then asked them to consider the two trays of brownies, one large tray and one small tray and how we could also show half of each tray of brownies. Again, students told her to draw a line cutting the rectangle into two equal size pieces. This process was repeated for the number lines to represent half way across the number line. Students were asked to think again how in each representation, one-half shows different sizes and how this could be true. After some time to confer with their peers, students shared some great insights. One student said that it matters what the ‘whole’ is. Splitting something in half means you make two equal pieces. Many students agreed and gave their classmate the ‘me too’ sign. In essence, one-half of one object can be larger than one-half of another object if their respective 'whole' is larger.

While we were getting close to the end of our time, I felt like students could use time the next day to not only revisit this discussion, but to consider some other one-half relationships in the Cuisenaire rods. I planned to have students start by exploring with the rods again and this time, finding any rods that showed a relationship where one rod was one-half another rod. This would allow students to apply their learning from today and use language to articulate how we name fractions. I’ll save the incredible findings that students had for another time!

I first want to praise the authors of ** Beyond Pizzas and Pies** for setting up experiences where students are thinking about big ideas around fractions. The questions that are asked and the flow of one experience to another builds to let students make sense of the mathematics. This lesson has been a proven winner that engages students from start to finish and leaves them wanting more.

Secondly, I think the Cuisenaire rod is such an underused manipulative. While they seem so simple with their bright colors and different lengths, their uses are endless. I especially love this tool when learning about fractions because they aren’t labeled. It requires the learner to notice the relationship between the pieces. Not only were Mrs. Jones' students using the rods to demonstrate what they thought as they shared their ideas with the class, but the tool served as a way to think through the math.

Take care,

Holly

McNamera, Julie and Shaughnessy, Meghan. ** Beyond Pizzas and Pies**. Portsmouth, NH. Heinemann. 2022

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