#### Day 1 Clip 1

In this clip we join Ms. Coombs as she works with the students on exploring graphs of the functions y=x2 and y=x3 near x=0.

#### Day 1 Clip 2

To further explore the “flat” behavior near the origin of the monomial functions, Ms. Coombs directs the students to consider y=x6. The function of this graph, in a standard window, looks so flat near the origin that the x-axis and the function become indistinguishable.

#### Day 1 Interview Clip 1

Following the classroom instruction Ms. Coombs and Pat meet to discuss the lesson.

#### Day 1 Interview Clip 2

In a second clip from the post-class meeting, Pat explains to Ms. Coombs what he means by having the students think about moving slowly across the domain.

#### Day 3 Clip 1

Ms. Coombs tries to use Pat’s ideas and directs the students to take baby steps to construct a graph that results from adding the function y=-x and y=x2.

#### Day 3 Clip 2

As the class proceeds, students continue to take “baby steps” across the x-axis and estimate the specific values of the functions to find the sum of the two functions.

#### Day 3 Clip 3

The procedure of step-by-step estimations that Ms. Coombs began in the previous clip continued on for three minutes. In this clip she begins to employ a form of covariational reasoning to help construct the graph of the sum of the functions.

#### Day 3 Interview Clip 1

In the after class meeting, Pat discusses with Ms. Coombs how precise numerical values are extra details. The students can, in fact, focus on estimated magnitudes to help construct the graph of the sum.

#### Day 4 Clip 1

Ms. Coombs employs Pat’s idea of using a graph without scales to help students focus on the important aspects of what it means to add functions.

#### Day 4 Clip 2

After students have completed their worksheet, Ms. Coombs guides a discussion of finding the sum of the two functions using an unmarked ruler to estimate magnitudes.

#### Day 5 Clip 1

On this clip, Ms. Coombs continues to generate the sum of the two given functions.

#### Day 6 Clip 1

After having time to work on the task, Casey, one of the students, places his work on the board. In this clip, Ms. Coombs directs a classroom discussion to review Casey’s work. She focuses on particular intervals and landmark points on the graph.

#### Day 6 Clip 2

In this clip we see Ms. Coombs continue to work the task dealing with how populations changed over time.

#### Day 6 Clip 3

In this clip we see Ms. Coombs continue to work the task dealing with how populations changed over time.