*(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.

## Is bad context worse than no context?

In elementary classes we consider it a good thing to be able to move from the abstract to the concrete. We ask students to count and perform arithmetic on objects, even contrived ones. We ask students to group socks, slice pizzas, and describe snowballs. A critical person might suggest these are all examples of pseudo-context, and they’d be right! These are more-or-less contrived scenarios that don’t

reallyrequire the context to get at the math involved. Why do we provide such seemingly inessential context? I’m venturing a little far away from my area of expertise here, but I’d guess it’s because it helps kids understand the math concept to have a concrete model of that concept in their heads.My question is this: in secondary classrooms, is there inherent value in linking an abstract concept to an actualized context? Even if the context is contrived?

I mean, yeah that’s bad. Comically and tragically bad. It doesn’t do anything to enhance understanding. I’d say the context actually

hindersunderstanding.The thickness of an ice sculpture dragon’s wing?That’s about three bridges too far.But what about a slightly less convoluted, but also-contrived, example. Say:

This problem is still most certainly contrived: dimensions of tanks aren’t often given in terms of

x. I’m not even suggesting this problem will engender immediate, massive engagement, but it might help students create a mental model of what’s going on with a third degree polynomial. Or at least the context allows students to affix understanding of the x- and y-axes once they create the graph of the volume of this tank.We provide similarly pseudo-contextual in elementary classrooms in order to enhance understanding of arithmetic and geometry.

From Burns’ “About Teaching Mathematics”

Of course, we also compliment such problems with manipulatives, games, play, and discourse, which secondary math classrooms often lack. In the best elementary classrooms we don’t

justprovide students with that single task. We provide others, in addition to the pure abstract tasks such as puzzles or number talks.Perhaps the true sin of pseudo-context is that it can be the prevailing task model, rather than one tool in a teacher’s task toolbox. In secondary math classrooms pseudo-contextual problems are offered as

themotivation for the math, instead of exercises to create models and nothing more.(See also: Michael’s blog post on Context and Modeling)