*(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.

I like the concept! I like the idea that each student’s answers can be a little different without being wrong (multiple correct ways to answer the problem). I like the comparison with peers. I like the follow up challenge (smallest area around the skeleton — though this brings up the question of whether the area has to be rectangular).

However, as a science teacher (as well as the wife of an archaeologist) I have one big beef — digging up dinosaur skeletons is not archaeology! I am uncomfortable seeing this common misconception repeated in an educational activity. Archaeology is the study of past humans. The study of ancient fossils is paleontology. When my husband talks to student groups, he is commonly asked about dinosaurs that he has dug up, and he has to spend time dispelling this myth (and explaining what it is that archaeologists do). It may seem like a trivial distinction to some, but I think getting the terminology right for a school activity is important.

Thanks for sharing your activity. I may make use of it in my own Algebra class — after updating the title!

Huh. Learn something new everyday. I’ll change the title then.

Nice! I like how the openness of this task and its whimsical-ness (really – the unicorn is perfect). The different solutions allow for a variety of possible conversations to deepen understandings, as you mentioned, about slope and area. Hoping to try this out next week. Thanks for sharing.

Cool activity. But… what *is* the notation for doing those intersections in Desmos? I’ve tried but can’t figure it out. When I type “and” it just wants to put in sliders for a, n, and d. Could you share Dan’s wisdom with the rest of us? Thanks!

You can snag the notation here: https://www.desmos.com/calculator/6v0rb28bsl

Got it, thanks! Now that I see that you can put in multiple restrictions, I’d probably encourage students to go with something more specific to each skeleton, like y>-30 (and then the restrictions) or x>20, but for a one-size-fits-all approach I guess x^2>=0 will always work. I might try this with some 8th graders this week 🙂