Systems of Linear Inequalities: Paleontological Dig

(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)

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  • 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:
  • 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. (

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


U-haul Linear Systems problem (updated and improved)

A couple weeks ago I posted this problem. I like the problem, but I wasn’t a huge fan of its solution and frankly, it sort of got away from me. Thankfully, my colleagues are more adept than I am at developing clear problems. So here’s the updated, improved version.


Moving On Up

Artifact / Entry Event

Facilitator : “Great news everybody. Principal ________ gave me a huge raise. It turns out that, contrary to all the media reports, the school district is flush with cash. As such, I’m moving on up. To the East side. I bought a MTV-cribs-like house. Unfortunately, I don’t have my hands on the huge sums of money yet, so I’ll be using a U-Haul to transport my currently meager possessions. Here’s a map of my old house, new house, and U-Haul rental location.”

[Facilitator shows slide / handout of map]

Facilitator: “Now, the local U-Haul rental has different fees for different sized trucks. The larger the moving truck, the more it costs. Here is their pricing scheme.”

[Facilitator shows slide / handout of U-Haul pricing table. Note: alternatively, the facilitator could direct them to the U-Haul website to obtain the specs: 14’ truck and 20’ truck. Note: there is also a 17’ truck for an up front cost of $29.95.]

Truck length U-Haul listed capacity Up front cost (1 day rental) Additional cost per mile
14’ 733 cu. Ft. $19.95 $0.79
20’ 1015 cu. Ft. $39.95 $0.79

Facilitator: “Now, I’m not exactly sure how much stuff I have. It might take 1 trip with these trucks, it might take 10. So I’m going to go calculate my furniture while you guys help me come up with a mathematical model, suggesting which truck I should get depending on the number of trips I have to take.”

Potential solution:

Students should first come up with a basic functional representation of the cost vs. mileage before applying it to the number of trips.

Cost = (up front cost of truck)+$0.79*mileage

, where m=mileage driven. This can be graphed as well.

To make a choice on which truck to get, students will have to determine a distance to and from each location on the map.

From U-Haul rental to Old Home: ~5.5 mi

From Old Home to New Home: ~8.5 mi

From New Home to U-Haul Rental: ~9 mi

Students then will need to set up a model in the form of a linear function, depending on the number of trips between the Old Home and New Home. Each extra trip after the first one will tack on 8.5+8.5=17 miles to the total mileage.

Cost = (Up front cost of truck)+$0.79*(mileage from U-Haul to Old Home + mileage between Old Home and New Home + mileage from New Home to U-Haul) + $0.79*2*(mileage from Old Home to New Home)*(number of trips – 1)

Cost = (Up front cost of truck)+$0.79*(5.5+8.5+9) + $0.79*2*(8.5)*(number of trips -1)

Cost = (Up front cost of truck)+$0.79*(23) + $0.79*2*(8.5)*(number of trips – 1)

Cost = (Up front cost of truck)+$18.17+$13.43*(number of trips – 1)

We can break that down for the three given truck sizes. For the smallest truck (the 14’ truck):

where x = # of trips to and from the New/Old Home

For the other two trucks:

Using a table:

# of trips:











14’ truck











20’ truck











Entry Event #2

Facilitator: “I’m still working on figuring out the number of trips I’ll have to take. I had several moving companies come over – before I decided to rent a U-Haul – and they each gave me a wildly different estimate of the amount of cubic footage of stuff I have.

Estimate 1: 2225 cu. ft
Estimate 2: 2891 cu. Ft.
Estimate 3: 3262 cu ft.

               Which size truck should I rent, based on each estimate?”

Potential solution.

Each estimate would entail a different number of trips based on the size of the truck rented. Recall,

Truck length U-Haul listed capacity
14’ 733 cu. Ft.
20’ 1015 cu. Ft.
Truck capacity

14’ truck (733 cu ft)

20’ truck (1015 cu ft)

Estimate 1: 2025

3 trips

2 trips

Estimate 2: 2891

4 trips

3 trips

Estimate 3: 3462

5 trips

4 trips

We can translate this into cost also using a table. (most cost effective)

Truck capacity

14’ truck (733 cu ft)

20’ truck (1015 cu ft)

Estimate 1: 2025



Estimate 2: 2891



Estimate 3: 3462



So for the three estimates, the first two would be more cost efficient using the 20’ truck while the last estimate would be more cost efficient using the 14’ truck.

Extension questions:

Can we develop a rule of thumb for differentiating between getting a 14’ vs. a 20’ truck (i.e. “If the # of trips using the 14’ truck is two or greater than the 20’ truck then it’s more cost efficient to get the 20’ truck.)?

Can we calculate the actual square footage of the 14’ and 17’ trucks? This incorporates volume standards.


Here’s a downloadable pdf version of the above activity.


I still am not 100% sure this activity is ready to go. I feel like it could still be improved. By all means, if you have any suggestions, let me know in the comments.

Can someone help me improve this problem? (U-Haul rental rates ; systems of linear functions)

So the Emergent Math family moved houses this weekend (editor’s note: they didn’t actually move the houses, they moved the furniture inside the houses). We were given basically three different options for U-Haul rentals of increasing size and increasing price. This led me to believe that a nice systems problem could be constructed from this.

I still feel that way, but it sort of got away from me. I think the idea of deciding whichtruck to rent + various mileage costs is pretty decent. But then things sort of spiraled out of my control and I ended up with a convoluted problem that was actually kind of tedious.

So I’m asking my faithful readership to help me out. I’ve split the problem up into two pieces – as I probably would have in a classroom setting. Help me make this problem better!

Basic gist of the problem:

Moving houses (ed. note: again, just the furniture – the author was not living in a double-wide). Three different U-Haul options of increasing square footage (733, 865, and 1015 cubic feet) with three different up-front costs for the U-Haul ($19.95, $29.95, and $39.95, respectively). Also, mileage costs $0.79 per mile – hence thenice linear functionality of the problem. Determine which U-Haul I should rent.

Documents (Part 1 and 2) below.

Best improvement to the problem gets a free oven mitt!