6 comments on “Systems of Linear Inequalities: Paleontological Dig”

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:  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.



6 comments on “Where inquiry and methods intersect”

Where inquiry and methods intersect

Had a nice, quick twitter conversation with Anna (@borschtwithanna) yesterday morning. Anna reached out with a question about providing methods in an inquiry-based classroom.

Anna was conflicted due to her students’ unwillingness to deviate from their inefficient problem-solving strategy. Rather than setting up an equation…

Setting aside for the moment that this is actually a pretty good problem to have (students willing to draw diagrams to solve a problem, even at the cost of “efficiency”), it does circle back to the age-old question when it comes to a classroom steeped in problem-solving: “Yeah, but when do I actually teach?”

The answer to that particular question is “um, kinda whenever you feel like you need to or want to?” The answer to Anna’s question is pretty interesting though, and I’d be curious what you think about it. Personally, I never had students that were so tied to drawing diagrams to solve a problem, that they weren’t willing to utilize my admittedly more prescriptive method. I do have a potential ideas though.

Consider Systems of Equations. This is a topic that is particularly subject to the “efficient” method vs. “leave me alone I know how to solve it” method spectrum. Substitution, elimination, and graphing were all methods that students “had” to know (I’ll let you use matrices if you’d like, I’m good with just these three for now).

Anyway, so I’m supposed to teach these three different methods for solving the same genus of problems. I want kids to know all three methods (generally), but also want to give them the agency to solve a problem according to their preferred method. Here are a few possibilities to tackle this after all three methods are demonstrated:

1) Matching: Which method is most efficient?

OK so matching is kind of my go-to for any and all things scaffolding. It’s my default mode of building conceptual understanding and sneaking in old material (and sometimes new material!).

In this activity students cut out and post which method they think would be the most “efficient.”

Students could probably define “efficient” in several ways, which is ok in my book. Also, it’ll necessitate they know the ins and outs of all three methods.

2) Error finding and samples of work

This is another go-to of mine. Either find or fabricate a sample of work and simply have students interpret. If you’re looking to pump up particular methods, consider a gallery walk of sorts featuring multiple different methods to solve a particular problem. The good folks at MARS utilize this in several of their formative assessment lessons. These are from their lesson on systems.

Screen Shot 2014-10-24 at 7.47.37 PM

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Students are asked to discuss samples of student work and synthesize the thinking demonstrated, potentially even to the point of criticism.

That’s a couple different ways to address methodology and processes that may turn out to be more efficient, while still allowing for some agency and inquiry on the part of the student.

What do you have?

6 comments on “Larry Ellison, billionaire CEO, makes unsound business decisions with regards to his basketball playing on his yacht.”

Larry Ellison, billionaire CEO, makes unsound business decisions with regards to his basketball playing on his yacht.

Larry Ellison, co-founder and CEO of Oracle, has gobs and gobs of money. How much money? Well enough that he can do this.


Boy that seems wasteful, doesn’t it. I mean, when I’m playing basketball on my yacht and I lose a ball into the ocean I just purchase an extra basketball. Wouldn’t it make more sense for Ellison to just buy a bunch of basketballs and grab a new one every time he loses one overboard? So my question is this: How many basketballs would Ellison have to lose in order to make the expense of basketball retrieval worthwhile?

Here is some of the board work we generated during the initial Problem Defining and Know/Need-to-Know process.


It’s critical that we understand our ultimate goal here: we want practice developing a mathematical model based on a given scenario. A model should, among other things, simplify a complex situation. We wound up focusing on only two variables: the cost of the annual salary and the cost of a basketball. A couple variables that folks tossed out ended up not being explored mathematically. As you may have experienced, when given a modeling scenario, students might throw out potential variables to tack on in perpetuity. There comes a tipping point where the mathematical model ceases to simplify a complex scenario and only confuses it further. I find this pretty typical of “make a budget” tasks or other accounting-type tasks (“what about sunscreen costs? what about health insurance? what about the yacht food? etc etc etc.”). When you’re facilitating the brainstorming process, I’d suggest you restrict the number of variables you’re including to two or three. This way, the entire class is focusing on the same few variables, keeping the focus on the model development, not the number of ingredients you can toss into the stew.


Here’s the initial PDF file if that embedded version looks goofy.

Basketball Overboard – Problem Solving Framework (pdf)

7 comments on “Evaluating energy efficiency claims”

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.

2 comments on “CNET has some TV viewing size/distance recommendations.”

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?

3 comments on “Why doesn’t Nike+ use math to encourage me to run?”

Why doesn’t Nike+ use math to encourage me to run?


The Nike+ app, which at the end of my run the other day, looked like this:

(editor’s note: yes, I’m slow. Thank you for noticing. Also, along with some encouragement in data format, I had Tim Tebow give me words of encouragement for bettering my pace.)

Now, there are a lot of numbers here, but I’m primarily interested in that last piece.

“You ran 0.10 mi more and 0’56″/mi faster than the average of your past 7 runs”

Why did Nike+ choose my past 7 runs? Was there some sort of algorithm to maximize how good I feel about myself?

Sadly, I think not. Witness my previous run:

So it looks like it just takes your past 7 runs and compares your mean distance and pace. That’s not very good motivation, now is it Nike+? Can we improve (at least, in my opinion it would be an improvement) Nike+’s distance and pace comparison to help the runner feel better about his or her progress?

Guiding Questions

  • Would a different measure of central tendency lead to a different, and perhaps more encouraging, data capture?
  • Would averaging a different number of past runs lead to a different, and perhaps more encouraging data capture?
  • Could we write an IF…THEN or other type of algorithm to encourage the runner?

Suggested Activities

  • Give some runner data (either fabricated or authentically generated; shoot, you can use my data if you want) and ask students to describe after each run “what should the app say in order to give the runner a sense of accomplishment?”
  • Once students have done that with individual data points, have students sketch out an algorithm or decision tree.
  • Test that algorithm or decision tree against a new set of runner data.

  • Compare decision trees and algorithms to see who’s is the “positivest”(?).
  • Turn into algebraic expressions if you want, presumably to help out the coders.

There are few things more discouraging than seeing that I’m actually running slower than my seven previous runs averaged out. At least package the data so I don’t feel like I’m out of shape.