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.


Evaluating energy efficiency claims


This (or other) energy efficient light bulb package(s).

Energy Efficient Bulb 20-75 w

So many opportunities here, depending on how targeted you want to be. Or, if you prefer, what kind of problem you plan to facilitate. There’s a clear nod to systems of linear equations (when one compares the time of payoff). There’s also an opportunity for some simple, linear equation building: evaluate the truth behind the $44 claim.

I’m even thinking of a 101qs video in which a perplexed customer at a hardware store is comparing this light bulb, and, say, one of these, though, these existence of incandescent bulbs is probably not long for this world. And, being Easter, hardware stores are closed today (fun fact: also, retailers really don’t like it when you take photos and videos in their stores). But that brings up a whole other can of worms: how much energy will countries save by switching to energy efficient bulbs? Like I said, lets of ways to go about this, depending on whether you want to be targeted or more exploratory.

Suggested questions

  • Is that $44 claim reasonable or bogus when you compare it against a bulb that uses 75 watts?
  • How does this compare with other energy efficient bulbs at the old hardware store?
  • What would happen if you switched every bulb in your house/school/neighborhood to energy efficient ones?
  • How much does a kilowatt-hour cost in our town? And what exactly is a kilowatt-hour?

Potential Activities

  • Take some predictions: does $44 savings sound about right over 5 years? Is that too high? Too low?
  • Collect some data on how much your lights are actually on in your house.
  • Plot five years of bulb use and see what happens.
  • Go around your house and count the number of bulb outlets you have. That data may be nice to have on hand.
  • Tables, graphs, equations, the usual bit.

Potential Solutions

Not sure what electricity costs in your particular neck of the woods, but Planet Money suggests a US average of $0.12 per KW-hr. These 20 watt bulbs usually cost around $12 per bulb, give or take. So our function looks like:

cost=$12+(20 W)*(1 KW/1000 W)*($0.12/KW-hr)*hours

Incandescent bulbs go for about $2, and comparing with a 75 watt bulb, our graphs look like this.

I actually get a savings over 8000 hours of $42.8:

(2+75/1000×0.12x 8000)-($12+20/1000 x 0.12 x 8000). That doesn’t take into account replacing incandescent bulbs more often. You could potentially get all stepwise functions if you consider the, perhaps 1000-2000 hour lifespan of an incandescent bulb.

(note the slightly different guesstimations of numbers in the planning form)

Final Word. Pretty much anything involving energy efficiency is going to allow for some systems problems. It’s all about tradeoffs, with higher initial costs gradually replaced by energy savings. Water heaters, A/C Units, automobiles, window insulation, you get what you pay for.

CNET has some TV viewing size/distance recommendations.

Feels like there’s a similarity (and a lot of other stuff) type problem in here.


From CNET:

 In a perfect videophile world, you’d want to sit no closer than 1.5 times the screen’s diagonal measurement, and no farther than twice that measurement to the TV. For example, for a 50-inch TV, you’d sit between 75 and 100 inches (6.25 and 8.3 feet) from the screen. Many people are more comfortable sitting farther back than that, but of course the farther away you sit from a TV, the less immersive feeling it provides.

I’m wondering if you could pair this with Tim’s TV 3 Act problem. Perhaps even Brian’s Holiday Shopping problem. There’s honestly a lot of stuff going on here from CNET: proportion, distance, maybe even a system of equations or linear programming problem (what with the upper and lower bounds suggested above, then toss in cost constraints).

Update (6/12/17): CNET has apparently redirected the original article to a generic TV buying guide, so the above text is no longer viable. However, here’s something from The Home Cinema Guide.

A good rule of thumb is that the ideal viewing distance for a flat screen HDTV is between 1.5 and 3 times the diagonal size of the screen – and we can use this to calculate both approaches.

Still, the work for the rest of this article reflects CNET’s original viewing recommendations.

Guiding Questions

  • How big a TV should I buy based on the above guidelines and my particular living room?
  • Could we develop a mathematical model to illustrate these guidelines? With, like, variables and stuff?
  • Alternatively, how could I set up my living room in order to fit the kind of TV I purchased?

Suggested Activities

  • Have students develop a model (or “rules” to follow) to express the above recommendation mathematically. (This one’s partially answered below)
  • Students could optimize viewing experience given a floorplan and a TV.
  • A Consumer Reports-ish type TV buying guide? We’re veering here…

Attempted Solution

So the initial model for the constraints listed by CNET aren’t terribly complex.

Constraint 1) “you’d want to sit no closer than 1.5 times the screen’s diagonal measurement”

Constraint 2) “no farther than twice that measurement to the TV”

So lower bound: d>1.5x ; upper bound: d<2x ; and there you have it.

Surely we could ramp up the complexity of the problem with some of the above floorplanning activities and additional cost constraints. How would you modify this situation to serve our mathematical purpose here?

Underground parking spots cost $30000-$50000 to build in D.C. This smells like a systems problem.


The Washington City Paper has a (rather lengthy) post on parking in D.C. Fair warning: it’s pretty wonky with zoning rules, ordinances, etc. However, the numbers caught my eye:

An underground parking spot costs between $30,000 and $50,000 to build, and residents pay for it one way or another.

“Let’s say it’s $100 per month. If you built the parking space, and it cost you $40,000, $1,200 per year doesn’t cover it,” says Four Points Development’s Stan Voudrie.”


On the supply side of the parking problem, costs are fixed: You can’t dig a hole and line it with concrete on the cheap. Demand is more dynamic, and to some extent, it responds to price. Unlike in suburban areas, most District landlords don’t pair spaces with the unit, which means that tenants pay between $100 and $300 more per month for their cars.

As these numbers appear, I suppose it’s more of a simple equation problem, but I feel like we could easily transform this into a nice, in-depth linear systems problem. For example you could contrive or find the cost of a garage parking spot and monthly fees, and figure out which one will pay itself off faster.

Basically, I’m just excited to finally find a potential systems problem that doesn’t involve cell phone plans.

Guiding Questions

  • How much do underground parking spots cost in our area?
  • How much do parking garage spots cost?
  • And what are the monthly costs of renting a spot?
  • How long would it take to pay off each spot?

Suggested Activities

  • Depending on how in-depth you want to get into this, you could turn this into some sort of apartment building project, or you could restrict it to a few-day investigation, having student find when each type of spot will pay for itself.
  • Again, if you really want to dig into this, you could ask student to develop a pricing system for their parking spots (i.e. would underground spots be worth more? what about spots closer in?).

Aside: this is like my fourth post on traffic or parking and mathematics. I’m starting to wonder if PBL could stand for “Parking Based Learning”.

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!