*(Editor’s note: the original post and activity mistook Paleontology for Archaeology. Archaeology is the study of human made fossils; paleontology is the study of dinosaur remains. The terminology has since been corrected and updated. Thanks to the commenters for the newfound knowledge.) *

Here’s an activity on systems of inequalities that teaches or reinforces the following concepts:

- Systems of Linear Equations
- Linear Inequalities
- Systems of Linear Inequalities
- Properties of Parallel and Perpendicular Slopes (depending on the equations chosen)

In this task students are asked to design four equations that would “box in” skeletons, as in a paleontological dig.

DOC version: (paleo-dig)

===

**Facilitation**

- Give students the entry event and instructions. Have one student read through it aloud while others follow along.
- Consider getting started on the first one (Unicorn) as a class. Should our goal be to make a really large enclosed area or a smaller one?
- Students may wish to start by sketching the equations first, others may chose to identify crucial points. Answers will vary.
- If you have access to technology, you may wish to have students work on this is Desmos. Personally, I prefer pencil and paper. Here’s the blank graph in Desmos: https://www.desmos.com/calculator/y1qkrfnsw2
- For students struggling with various aspects of the problem , consider hosting a workshop on the following:
- Creating an equation given a line on a graph
- Finding a solution to a system of equations

- Sensemaking:
- Did students use parallel and perpendicular lines? If so, consider bubbling that up to discuss slopes.
- Who thinks they have the smallest area enclosed? What makes them think that? Is there any way we can find out?
- Let’s say we wanted to represent the enclosed area. We would use a system of linear inequalities. Function notation might be helpful here:
- f(x) < y < g(x) and h(x) < y < j(x) (special thanks to Dan for helping me figure this notation out in Desmos!)

=== Paleontological Dig ===

Congratulations! You’ve been assigned to an paleontological dig to dig up three ancient skeletons. Thanks to our fancy paleontology dig equipment, we’ve been able to map out where the skeletons are.

**Your Task**: For **each** skeleton, sketch and write *four* linear functions that would __surround the skeleton__, so we may then excavate it.

**Check with your peers**: Once you have it, compare your functions to your neighbors. Their answers will probably be different. What do you like about their answers?

**Optional**: For the technologically inclined, you may wish to use Desmos. (https://www.desmos.com/calculator/y1qkrfnsw2)

**Challenge**: What’s the smallest area you can make with the four functions that still surround each skeleton.