Using chocolate to teach calculus

Tim Chartier, an Associate Professor of Mathematics at Davidson College, has a series of ingenious and delicious methods for illustrating fundamental concepts from calculus using chocolates. I had a brilliant calculus teacher, Alvin Anson, but I think a little chocolate would have improved even his techniques:

Where’s the chocolate? Rather than shading a square, we will place a milk chocolate chip on a square we would have colored red and a white chocolate chip on a region that would have been white. To begin, the six by six grid on the left becomes the chocolate chip mosaic we see on the right, which uses 14 white chocolate of the total 36 chips. So, our estimate of π is 2.4444. We are off by about 0.697.

Next, we move to an 11 by 11 grid of chocolate chips. If you count carefully, we use 83 milk chocolate chips of the 121 total. This gives us an estimate of 2.7438 for π, which correlates to an error of about 0.378.

Finally, with the help of public school teachers in my seminar Math through Popular Culture for the Charlotte Teachers Institute, we placed chocolate chips on a 54 by 54 grid. In the end, we used 2232 milk chocolate chips giving an estimate of 3.0617 having an error of 0.0799.

(via Neatorama)



  1. it’s not monte carlo since it’s not random. the phrase you’re looking for is “discrete approximation”.

    as a surly academic, i want to hate this, but i really can’t. it’s a decent demonstration of taking limits and it would probably get some attention which otherwise would have been dissipated. if you had a lot of time and chocolates of various sizes, you could even hint at fractal dimension.

  2. Personally, I don’t like the definition that a square with ANY part of the curve is given to white chocolate.  I feel that the dark chocolate should have those.  Yes, I am in favor of Manifest Destiny, but ONLY for dark chocolate.

  3. Oh no, that chocolate would never remain on the board unmolested long enough to work out any calculus problems.   ‘Where’s the chocolate?’  Indeed.

  4. It is cute, but it isn’t calculus.  I can’t think of a good use of chocolate for teaching calculus, except as an incentive.  However, this chocolate idea is great for teaching both geometry and algebra.  You can derive  formulas for the areas of various geometric shapes such as squares, rectangles,  rhombi, trapezoids, and triangles using chocolate chips.  If you are brave you could derive formulas for  some 3D shapes such as pyramids.  You can also derive algebraic formulas such as the sum of the first k integers equals k(k+1)/2 and x^2-1=(x+1)(x-1) by rearranging chocolate chips.  There is a wonderful book called “Proofs Without Words” available from the American Math Association.  Many of its proofs could be done with chocolate.

  5. I tried this method out, but I seem to encounter an error whereby my
    calculations are increasingly off by about 1 every 30 seconds.


    Wonder why that is?


  6. The approximation would be much more accurate if we counted the tiles that have an apparent majority of their area inside the curve.

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