Intro: How Many Grapes Equals a Dozen Peaches?
My 6-year-old cousin quipped to me that 100 grapes is equal to a dozen peaches. Deciding he was referring to volume, we came up with the following experiment which uses the displacement of water to equate volumes. Though simple (all materials in photo), this highly visual and surprising experiment has many important lessons and can easily be adjusted to be appropriate for any grade 1-12.
Fill a glass measuring cup with enough water to submerge a peach. If you do not have a measuring cup, you may simply use a big glass jar and a dry-erase marker to mark the water levels. Or, you may simply use two identical class jars side-by-side filled to the same level of water.
Have your students note the water level (300 ml in this photo).
Place the peach in the water and have your students note the change in water level (to 450 ml in this photo). Really, this experiment can be done with anything, but I like round objects (because of easy volumetric calculations) and fruit since its fun to eat -- although many are too buoyant to do the experiment properly since it must be submerged (or nearly so) to get an accurate volume measure.
Ask the students why the peach must be submerged below the water line in order to get an accurate measurement of volume.
For students 6th grade and up, you might consider discussing the relationship Density = Mass / Volume and which property is most relevant to this experiment. For students 9th grade and up you may even have them calculate the density of the peach and compare it to the density of water.
Remove the peach but allow most of water to drip off it -- if you want to be precise (as in a chemistry class), you could add the precise water necessary to return it to its starting level.
Before beginning this step, ask your students to make predictions about how many grapes will be necessary to displace an equal amount of water (to raise the water to the same line as the peach)-- they will probably be surprised (and you may be too!).
With advanced students (9th and up), you might guide their predictions -- have them make a guess (or measure) how many times larger the radius of the peach is to the grape. Leave it at that, hoping that some will remember the radius is cubed in calculating the volume of spheres (4/3*pi*r^3), and thus the peach having roughly 3 times the radius of a grape corresponds to about 3^3 = 27 times the volume of the grape.
Add grapes one at a time to the jar until you reach the same water level as the peach (450 ml in this photo.)
Remove the grapes and count them, in this case it was 32 grapes!
With my first grade cousin, I arranged them in a grid as shown to introduce him to the concept of multiplication. This is also advisable with 2nd and 3rd graders. With 3rd grade and up, you would have them do the final calculation of 12 x 32 = 384 to arrive at the number of grapes to a dozen peaches (volumetrically.)
For advanced students:
- You might discuss sample bias, as in the picture you can see many grapes on the vine are very much smaller, which I ignored for the simple version. In this case, I would split the room into groups and not even mention that they should consider it, but have each group decide how to proceed with the experiment. Afterwards, they would have to justify how and why they chose the grapes they did.
- To me, the most important lesson from this experiment (and one which many adults may have forgotten) is that volume goes by the cube to the radius -- that although the radius is only 3x bigger, the volume is closer to 30x bigger. So, as in much of science, looks and intuition can be deceiving.
- In a class in which d= m/v is just being introduced, this would make a nice starting point. From here, you might explain that with this simple equation your students can now calculate the mass of a giant boulder just by finding the properties of a little pebble chipped off of it!
Feast on the delicious fruit (don't forget to share)!