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Exponential (and tooth) decay

Sunday 17 January 2016

Ever decreasing m and ms

Just a quick post to remind everyone of this classic activity. I first saw it on a workshop with Jane Forrest and new it would be a winner. It is a wonderfully simple way to look at modelling exponential decay.


In short

  • Get a large amount of m and ms,
  • Pour them out on a tray, count them and then eat all of the ones that have the 'm' showing upwards.
  • Then shake them all about, count them and eat all the ones with the 'm' showing upwards and so on.
  • Keep going until they are all gone and keep a record of the number of m and ms left after each 'shake'

Model

With the flatter, chocolate m and m's, we can predict that the amount left will reduce by approximately 50% each time. The less regular peanut m and m's are less predictable. In any case, the small amount of data is very easy to enter using a tool like Desmos. Then the key is to ask students to think about the features of the function. Start with the general expression....

begin mathsize 24px style f left parenthesis x right parenthesis equals k cross times a to the power of x plus c end style

You may have already established in previous activities that the y intercept of the function is given by k + c, whilst the asymptote is at y = c. Students can then use the features of their data to fit a model, that they can adjust.

From an 'understanding' point of view, this is infintiely preferable to an 'auto fit' function that many packages offer.

Once the technique has been practiced it can be used on other, more complex exaples such as those here in the Modelling World Population Growth activity.

Short, sweet (literally) and powerful!



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