In this activity, students familiarize themselves with the context of Nazca lines, which will be used throughout the lesson. As students consider these large geometric objects and their scale, they are making sense of problems (MP1). Students begin by reading or researching about the Nazca lines in order to familiarize themselves with the context.

Invite several partners to share one question with the class, and record their responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “length,” “perimeter,” “area,” and “slope.”

This activity gives students an opportunity to apply their work with figures in a coordinate plane to real-world situations. This context also invites students to use the coordinates to make sense of the scale of the figure. Students reason abstractly about the figure in the coordinate plane, then interpret their calculations as they make sense of the scale and quantities involved (MP2).

Ask students, "How big is this triangle?" Invite them to compare the length of 846.5 meters to other lengths to get a sense for how large it is. (It is more than half a mile, it is longer than the Burj Khalifa is tall, it is more than twice around a running track.) Invite them to compare the area of 13,630 square meters to other areas as well. (It's more than 3 acres, or about 140,000 square feet. You could fit 70 2,000-square-foot houses inside!)

Invite students to share how they might create a right triangle that is that large without using modern tools. 

This activity gives students the opportunity to consider mathematical questions about the Nazca lines and figures, and to then solve them. Students engage in the modeling process as they consider problems of interest, then determine how to represent the problem, perform the necessary calculations, and interpret their findings (MP4). 

Invite groups or students to share the questions they came up with and their solution strategies. Highlight language that students use to describe their strategies, including “slope,” “point-slope form,” “altitude,” and “distance.”

If time allows after student sharing, here are some additional questions for discussion:

• How could we find the area of the triangle? (We need to multiply base times height. Since it wasn't a right triangle, we had to figure out the height. We could use point-slope form to find an equation for the line that the altitude is on and the line the base is on, then find their point of intersection. We can use the Pythagorean Theorem or distance formula to find the length of the altitude and the base, and then use those values to find the area of the triangle.)
• How could we categorize the quadrilateral? (We can see that it isn't a square or a rhombus, because one pair of sides is much longer than the other. We can use the slopes to see if opposite sides are parallel or if adjacent sides are perpendicular. The slopes of the opposite sides are very close to being the same, so it is approximately a parallelogram. We can also see that the slopes of the adjacent sides are close to being opposite reciprocals, so it is approximately a rectangle. Since the long sides are much closer to being parallel than are the short sides, it might also be considered a trapezoid.)

Invite students to share their observations about the size of these figures and how that connects to their understanding of the tools available 2,000 years ago.

Math Community

Invite 2–3 students to share the norm they chose and how it supported the work of the group or to share a realization they had about a norm that would have worked better in this situation. Provide these sentence frames to help students organize their thoughts in a clear, precise way:

• “I picked the norm ‘_____.’ It really helped me/my group because _____.”

• “I picked the norm ‘_____.’ During the activity, I realized that the norm ‘_____’ would be a better focus because _____.”