Math Class Drops the Mic

A blog about teaching, with an emphasis on math.

Saturday, October 10, 2015

The Shepherd's Calculation

From the balcony of my new apartment I can see the Mediterranean Sea. This is what it looks like:

I live in Spain, where I’ve taught math at international schools for the past four years. I taught and surfed in southern California for many sunny seasons, and last week I discovered that you can surf in the Mediterranean too! Can you tell which photo below depicts surfing in California and which surfing on the Mediterranean?

That was a trick question actually—I promise no more trick questions in this post, but I couldn’t resist: They are both pictures of surfing in the Mediterranean! I still can’t believe there are waves here. And here’s another surprising revelation, in photo form, from my balcony:

So, there are shepherds here! Well, primarily he’s a goatherd, to be honest. His name’s Claudio. Here he is with his assistant:

I really wanted to talk to this guy, because I couldn’t imagine there was sufficient remuneration to be gained from these ruminants! How much could he possibly make? Well, I won’t reveal Claudio’s current economic situation, but from talking with him, here’s a super-simplified version of how he milks this gig for all it’s worth:

He gets about 3 liters of milk per day per goat, and 2 liters per sheep. That’s is a lot of milk! Cows in modern dairies produce about 25 liters a day now, which is so mind boggling that I dairyn’t consider it further! Claudio and his big family can only milk about 400 liters a day, absolute maximum.

The sheep are a bit more difficult to manage, and they are bigger. Goats take up one “stall” in the barn, but each big sheep takes up two! He’s got 300 stalls. (It’s more complicated than this, but go with it!)

He used to make about 1 for each liter of goat’s milk, and 2 for each liter of sheep’s milk, because the sheep’s milk is richer. Not punning there, it’s apparently creamier!

So, naturally, let’s make some inequalities and equations!

Of course, we can graph the inequalities. Perhaps you recognize this as a classic linear programming problem!

Here’s the graph, which shows us all the possible combinations of goats (x) and sheep (y) that meet Claudio’s constraints, in the region that is double-shaded. For example, clearly 40 goats and 50 sheep is possible for Claudio, because (40,50) is in the double-shaded region. (Note: Even though goats can be stubborn, their numbers are never negative!)

Finding the maximum revenue requires finding the corner points of this region and plugging them into the Revenue equation to see which combination returns the most money. Why the corner points? Below is what happens when you graph the revenue equation for different revenues. Notice that the revenue line doesn’t cross into Claudio’s world until we hit a revenue of about 650.

The combination of goats and sheep where the revenue line enters Claudio’s world is about 50 goats and 125 sheep. I leave it to you to determine the exact answers algebraically!

You might have noticed that above I said he used to make 1 per liter of goat milk and 2 per liter of sheep milk. Nowadays, the milk market has dried up, and he makes only 0.5 per liter of each kind of milk. If you run the numbers again you’ll see the reason why he seems to have more goats than sheep now in the picture. He and I didn’t talk about this, but it seems like the only reason he still has sheep is that he had them before, and you never know when the market might shift. Among the many realities complicating all this is that his local milk market isn’t as liquid as you might think—he only sells to one buyer and doesn’t always sell all his product.

Nevertheless, Claudio and his family have been doing this difficult work for generations and Claudio seems relatively content. For my part, I feel lucky to have visited with someone living a truly pastoral life right next door, and I hope that you enjoyed getting some insights into the economics which drive the animal drivers! I’ll eventually put this into a Mission on my Teachers Pay Teachers store (sooner rather than later upon request). Look for more real-world math at my TpT store, Courage To Core, and give the kids something to graze on! (Click the kid below!)

And for more intriguing posts about math in real-life, click below! Dart games, fish tanks, tasty tarts and much more!


  1. Great post! I love how you tied math into real life. Providing a connection to the material, and keeping things interesting definitely fosters an interest in math. I am extraordinarily jealous and LOVE the view from your apartment. My heart will always belong to the sea.

    1. Thanks for the enthusiastic response! It was definitely s stroke of luck that I saw this guy and immediately the question popped into my head--how does that work exactly? And he was super friendly and open too. It was funny though, he definitely spends a lot of time with the animals because he spent half the time explaining the particular ages, health statuses, milk production rates and so on of each goat that wandered up. He definitely knows his flock!