7^{th} graders are sorting decimal, fraction, and picture cards based on whether they were equivalent to each other. This group was wrestling with whether 0.6 and 0.60 were equivalent decimals. Erica specifically included cards like 0.6 and 0.60 to make visible students' ideas about place value.

**Focus Practice:** Argumentation (MP3)

**Special Note about Erica’s Background**: Erica filmed this lesson as part of her work as a coach for elementary mathematics teachers.

To reach consensus, students engaged in argumentation, characterized by students sharing, justifying, evaluating, and responding to one other’s ideas. To facilitate a classroom environment where students construct knowledge with each another, Erica has *sentence frames* taped to each student’s desk. See links to related materials for more information about how to design card sorting tasks.

1. What reasons did Jamie provide for why 0.6 was equivalent to 0.60?

2. Why was Jamie ultimately successful in persuading the others in her group that the two decimals were equivalent?

3. What strategies did Erica use to support her students in engaging in argumentation?

**Argumentation (MP3)**

One goal of argumentation is for students to reach consensus about a mathematical or scientific idea. This process involves students presenting claims, justifying their ideas with evidence, evaluating each idea, and responding to the ideas of others. Students often leverage the other mathematical practices in service of engaging in argumentation and developing a consensus understanding.

**Argumentation in Context**

In this group of four, two students are discussing whether 0.6 and 0.60 are equivalent decimals. Jamie and Elise state competing claims and then use evidence. To justify their arguments, both students attempt to use mathematical reasoning [MP2] involving ideas about place value [MP7].

**Understanding Jamie and Elise’s Ideas**

Jamie believed that 0.6 and 0.60 **were equivalent** because the 6 was in the tenths place for both decimals. Elise believed that they **were different** because there was a 0 in the hundredths place.

**Attempts to Persuade Elise**

To persuade Elise to revise her idea, Jamie noted that there is a non-visible 0 after the 6 in 0.6. She attempts to explain that writing an additional zero at the end of the decimal does not change the decimal’s meaning.

**Encouraging students to persevere**

Rather than continue with the argument, Elise wanted to move onto another set of cards where there might be more agreement. Upon hearing this, Erica encouraged them to persevere through the problem [MP1] to reach consensus. Erica polled the entire group to understand each student’s opinion; everyone except Jamie believed that decimals were not equivalent. Upon hearing the results, Jamie stated, “Three against one, the judge has spoken.” In response, Erica questioned the students’ standard for consensus (“Is that how we come to an agreement?”) and encouraged Elise and Jamie to explain their competing claims using evidence.

**Argumentation Part 2**

Elise repeated her idea that the decimals were different. Rather than repeating the same reasoning, Jamie used an alternative approach [MP1] involving equivalence between decimals and percentages [MP2]. She noted how 0.6 and 0.60 were both equal to 60%. However, when the 0 was placed in front of the 6, the meaning of the fraction changed 60% to 6% (“There's nothing in the tenths spot so this would just be 6[%].”) In doing so, Jamie used her ideas about the relationship between decimals and percentages as an additional source of reasoning [MP2] to justify her argument that an ending 0 does not change the meaning of the decimal. This line of reasoning successfully persuaded the other group members to revise their idea.

Watch the classroom video from Erica’s perspective as she explains her decisions and highlights features and things for teachers to notice when supporting students’ engagement in argumentation.