Tricks for Counting Treats

There are lots of treats (including freebies) in this post! Read on...

The leaves are changing color in the Blue Ridge mountains of North Carolina. The surf is picking up at Pacific Beach in San Diego. The elk are tangling in the forests near Jackson Hole but soon the first snow will urge them to warmer plains. Great flocks of warblers are touching down and stirring the glassy waters of the salt marshes of Louisiana. A welcome reef of cloud dulls the Mediterranean sun in Alicante. Autumn.

Autumn. At high schools everywhere, the clubs and student governments and teams have gathered themselves and now they raise their voices in a full-throated Autumnal cry: Let the fundraising begin! One of my favorite fundraising efforts is the ever-enticing challenge to guess the number of candies in a jar. Of course, there is always the possibility that the candy conceals a couple tennis balls, but I always assume the jar is authentically full and attempt to calculate the number. It’s tough to do without access to the jar, but if you’ve got a ruler and can get the dimensions you’ve got something to go on.

 But then what? Then you go home and fill a much smaller jar with the same candies!

 Then calculate the volume of the small jar and the large jar. (I leave that to you!) and set up a proportion:

But there was clearly some imprecision in measuring in the interior dimensions of a not exactly cylindrical cylinder!  Also, my larger jar in this case was actually narrower, which might actually reduce the efficiency of the packing of the candies, which are pretty large.

Without trying to solve a packing problem, and assuming we couldn’t just read the volumes from the label or use a measuring cup, what other methods could we use? Perhaps we can add some tools. If we had an empty jar equal to the target jar and some water, we could find out how many small jars of liquid are required to fill the large one!

Of course, we are still relying on an imprecise measurement of the fraction of a small jar we used at the last pour. Can we do better? Let’s add a scale to the toolkit. Now, if we find the mass of water which fills each jar, we’ve got a perfect proxy for volume.

It turns out the the true number of candies was 95! And it comes as no surprise that the best solution came from a great proxy for volume. Not only is the mass a great proxy for volume, a gram of water has exactly a volume of 1 cubic centimeter.

I’ve created a freebie which you can use to guide your student through the same series of experiments. Check it out HERE, as well as all the abundant treats for you at the intriguing blogs of the innovative Teacher-Authors of Teachers Pay Teachers below! Happy Halloween!

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