5^{th} grade math teacher, Bob, approached coach, Jack, to express his concern that his students could “procedurally know how to change a fraction to an equivalent fraction” without understanding why they were equivalent. In response, Bob and Jack co-planned a lesson that focused on developing this understanding.

Bob asked the students to use their whiteboards to represent 2/4 and 4/8. Students then shared their representations with the class. To represent 4/8, Joshua constructed an 8 X 8 grid and shaded in half of the squares. Through small group and whole-class discussions, students shared their reasoning and engaged in argumentation to decide whether 32/64 was equivalent to 4/8.

**Focus Practices**: Reasoning (MP2) and Argumentation (MP3)

**Special Note: **Teachers were invited to observe these 5^{th} graders engaging in argumentation with each other so you will see a number of additional adults in the room.

1. What reasons did students provide for why 32/64 was or was not equivalent to 4/8?

2. Why did Jack and Bob ask the students to "turn to their elbow partners"?

3. How was the pizza metaphor used to help students make sense of equivalent fractions?

4. How did Jack and Bob respond to incorrect ideas? How might you have reacted similarly or differently?

## Abstract and Quantitative Reasoning (MP2)

During this clip students were engaged in abstract and quantitative reasoning to evaluate Joshua’s claim that 32/64 was equivalent to 4/8.

**Understanding Joshua’s approach**

To represent 4/8, Joshua created an 8 by 8 grid and shaded 32 out of the 64 cells (MP4). He used abstract and quantitative reasoning to explain how shading 32 squares in an 8x8 grid was equivalent to shading 1/2 as well as 4/8 of the squares.

**Giving students “time to process” and evaluate Joshua’s reasoning**

The students are the individuals responsible for understanding, analyzing, and evaluating Joshua’s reasoning. Bob and Jack’s students were given “time to process” the ideas raised in the discussion (“turn to your elbow partner”) and clarify their understanding of ideas in the class (“Would somebody clarify what he just said?”).

**Using the “Big One”**

Robert recognized how 4/8 could be made equivalent to 32/64 by multiplying both the numerator and denominator by 8. His partner, Corey, recognized how Robert’s approach was similar to their past use of the “Big 1”, which involves multiplying a fraction by a fraction where the numerators and denominators were the same (e.g. “8/8”), which is equivalent to “1”. Thus, Corey connected the current situation with concepts or skills previously learned in class (MP1).

**Use of modeling to mediate student reasoning**

When evaluating Joshua’s ideas, Nathan and his partner Lorenzo leveraged two different type of reasoning. The first involved simplifying the fractions to show their equivalence. The second involved using pizza to understand mathematical concepts using connections to the real world (MP4). In his argument, Nathan noted that a 64-slice pizza could be bigger than an 8-slice pizza. In this case, eating half of each different-sized pizza would not result in eating the same amount of pizza. Nathan is considering the size of the referent whole. In doing this he is applying his knowledge of the real world (cutting most pizzas into 64 slices would be silly) to this mathematical situation, and this consideration is leading him to reflect on his understanding of the equivalence between 4/8 and 32/64.

**Understanding Nathan's Idea**

Will and Jen took steps to *clarify* their understanding of Nathan’s idea. Will noted how slices from the larger, 64-slice pizza would result in people getting “[filled-up] a lot faster” than if they were to eat 4 slices of a smaller, 8-slice pizza. Jen reasoned that a 64-slice pizza might have to be bigger because it could be “harder” to cut a pizza into 64 “really tiny slices”. In both cases, Will and Jen did not evaluate Nathan’s idea. They simply made public their understanding of and plausibility for Nathan’s idea.

**“Laser-cutting Pizza”**

To help students focus on *fraction* equivalence, rather than the difficulty of slicing pizzas into small pieces, Jack asked the students to consider two identically-sized pizzas that could be cut using a “super ninja laser pizza cutter.” It’s important to note that the teachers did not give up on the pizza analogy because the students considered it as a useful way to think about this mathematical concept (MP1 and MP4). By the end of the episode, most students agreed that eating 4/8 and 32/64 of the same pizza would result in eating the same amount of pizza (“they’re still going to be filled up a the same amount because it’s the same pizza – it’s just split into smaller pieces”).

## Argumentation (MP3)

**Attention to alternative arguments**

In this class discussion, students evaluated the Joshua’s justification for why 32/64 was equivalent to 4/8 and 1/2. However, some students, like Nathan, considered reasons for why the fractions **would not be** equivalent, which raised issues about the pizza size.

**Time to process**

Before each instance of idea evaluation, the students were given the opportunity to process their own understanding and assess their level of agreement with the presented ideas. This “time to process” supported students’ engagement in argumentation because the students could better explain and defend their arguments in a public setting.

For students to be able to respond to, defend, or rebut an idea, it’s important that students have an understanding of the target idea. When new ideas were raised, the students spent time trying to understand Nathan’s idea (“Would somebody clarify what he just said?”) and consider Nathan’s reasoning for why a 64-slice pizza might need to be bigger than an 8-slice pizza. In doing so, students had to attend to their classmates’ thinking and determine whether they believed the reasoning to be sound.

**Teacher did not correct student idea**

It’s important to note the teachers did not indicate that Nathan’s answer was incorrect. Recall that one of the original goals was for **students** to derive a *conceptual* rather than a *procedural* (e.g. simplification or multiplying by the “big 1”) understanding of fraction equivalence. Thus, the students had to be actively involved in understanding the reasoning of others, building on each other’s ideas (I agree or disagree with…), and engaging in argumentation.

Watch the classroom video through Jack’s perspective as he explains Bob and his decisions and highlights important features and things for teachers to notice when supporting student sensemaking.