Erica is a middle school math teacher and coach. The students are using graphs to model changes in the wild and captive condor population over time. Specifically, they are investigating the question, “When can we de-list the California condor from the endangered species list?”

**Focus Practice**: Modeling (MP4)

They realize that the population of condors had approached zero at one point in time and that to be “de-listed” the population would need to be at a much higher level. Students complete the following tasks in small groups sharing their findings with the class:

Task 1: During which period does the captive condor population increase about 20 birds in two years?

Task 2: When is the condor population increasing the most rapidly? Is it in the wild population or the captive population

Task 3: When is the conservation group more effective at repopulating the wild population?

1. What strategies do students use to analyze the changes in the condor population over time?

2. What are some of the benefits of students working in small groups and sharing their ideas with their classmates?

**Modeling (MP4)**

Modeling with mathematics involves students representing and making sense of something in the real world. In this case, students constructed a graph to study changes in the wild and captive condor populations (y-axis) over time (x-axis). The graph is a model because it **maps relationships **between important **quantities**. To interpret the model’s ideas, the students use abstract and quantitative reasoning (MP2).

**Task 1**: During which period does the captive condor population increase about 20 birds in two years?

This task was a “warm-up” to get the students oriented to the data. Initially Rohan and John focused on 20 years instead of 20 condors but self-corrected and went on to claim that the captive condor population increased by 20 condors between 1996 and 1998. To measure the population change, the students counted the number of “squares” and engaged in quantitative reasoning by realizing that each square represented 10 condors, justifying their claim, “It's 20 birds because each one is 10.”

**Task 2:** When is the condor population increasing the most rapidly? Is it in the wild population or the captive population?

**Small Group Work**

Matt and Sam drew vertical and horizontal lines to connect the two points to determine the relationship between the change in the condor population and the change in time. The students figured out for themselves that they needed to look for the line segment that had the steepest slope.

Rather than simply use visual inspection the students reasoned abstractly and quantitatively (MP2) by computing the slopes of two line segments (MP6). They assessed the reasonableness of their calculations by checking their answers with each other.

**Group Presentation**

After working in small groups, the students shared their findings with the whole class. Rob and Jenny found that the wild population changed the most rapidly between 2004 and 2013~~, ~~saying “128 [condors] over 9 [years]” which they claimed was greater than the other calculated slopes. Calculating the slopes for both wild and captive populations was critical for “changing [their] answer”. Maintaining an interval of 9 years for both populations enabled them to make the comparison between the two populations. This insight demonstrates how this pair engaged in mathematical reasoning and was able to justify their work.

**Task 3:** When is the conservation group more effective at repopulating the wild population?

Naomi determined that the conservation group was effective in repopulating the wild population.

She provided three types of mathematical reasoning to justify her answer. First, the captive population curve was “steeper” than the wild population curve. Second, she calculated the slopes, which were in the form of fractions (e.g. 110/9), found a common denominator by multiplying both numerator and denominator by 5, and compared them. Third, she noted that the final captive population was “below” and “less” than the wild population. It was unclear whether she meant that the captive population had a smaller population or was “less effective” than the wild population. Regardless of the interpretation, Naomi used multiple lines of reasoning to support her claim.

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 modeling.