Mathematics/Physics

Filling the Glass (Water, Air, and Fractions)

Objective(s):

Appropriate for elementary-grade students who have a beginning knowledge of fractions and measuring. Students will associate the physical act of measuring a fraction of a given amount of water with the arithmetic operations involved. Each group will use a different method of measuring, and compare/contrast the associated equations with those produced by other groups. The lesson is intended to integrate science and mathematics.

Materials Needed:

Class:

(Note: All glassware must be transparent.)

2 identical straight-sided, flat-bottomed glasses (e.g., "Tom Collins"-style), 2 identical sloped- or curved-sided glasses (e.g., "rocks" or champagne glasses), 12 identical plastic glasses (any style), 2 graduated cylinders, 2 graduated beakers, 1 meter stick or 12-inch ruler, 1 roll masking tape (optional), 1 marking pen (optional), 1 balance-beam scale with weights, at least 3 liters of water (if water source is not handy), 1 clear glass bowl, 1 chalkboard/whiteboard/overhead projector (for the sake of brevity, we will refer to this as the "board").

Each group:

1 calculator, pens/pencils and paper for each member.

Strategy:

Divide class into groups of 3 or 4 members each and distribute group materials. Place class materials together on a table accessible and visible to all, and close to chalkboard/whiteboard/overhead projector. Begin by filling one of the straight-sided glasses with water. Ask the class to describe the glass using the words "full" and/or "empty." Ask them to do the same with the other straight-sided glass (at this point, it is better to do this silently on their own paper, since we are not ready to discuss this yet). Now call up one group and challenge them to move half the water from the first glass into the second. Most will simply pour back and forth until the water in both glasses is at the same level. If they start to reach for the measuring devices, ask them to explain the procedure they had in mind, discuss its merits, and challenge them to do it instead without measuring devices. Once this is done, discuss the concept of taking half of something and write "1/2 of a glass of water" on the board. Once again, ask the class to describe (on paper) the two glasses using "full" and/or "empty."

Now empty one of the glasses and challenge another group to move half the water from the other glass into it, this time using the ruler. Most will measure along the side of the glass to the level of the water and calculate half of that length (some may wish to use the masking tape and marker to mark levels). Point this out and ask someone from the group to write an equation on the board that describes what they just did. When they have written "1/2 X (measured length) = (result), ask them what the measure of a full glass would be. Some will go back and measure, others may say, "Just multiply by two," or something to that effect. Once again, have them represent this mathematically on the board. For the last time, ask the class to describe (on paper) the two glasses using "full" or "empty." Set aside the glass with water in it for later.

Send all students back to their seats and go to the board. Point to the original phrase and ask which of the things they wrote could replace the words "of" and "a glass of water." Is this an equation? (No). Referring to the various things on the board, and adding others as you go, guide them through an introduction to multiplying fractions times fractions and fractions times whole numbers.

Replace the glasses with a sloped- or curved-sided pair. Call up a third group and challenge them to fill one glass half-full of water. If they do so by pouring back and forth, point out that the resulting level may not be where they might have expected, and discuss why. Then ask them to perform the same task using the balance-beam (if this was their first instinct, do the water-level discussion when they finish). Would the ruler have worked this time? (No). Ask them to represent the procedure mathematically on the board, using the previous examples as a guide. Then, as before, empty the water from one glass and ask them to transfer half the water from the other glass into it. Once again, ask them to represent the procedure mathematically on the board.

Send this group back and call up the fourth group. Have them repeat the above procedure with either the graduated cylinders or beakers, explaining the reasons for their choice. When they have finished (including the math), send them back and engage in a general discussion about choosing most appropriate forms of measurement (eyeballing, length, weight, volume, etc.) as well as measurement tools.

As a wrap-up, take out the two straight–sided glasses and fill one with water. Ask the class how they responded when asked to describe it as "full" or "empty." Do the same with the other glass. Most will have said, "empty" for this one. Fill the glass bowl, upend the glass in it, and ask, "If the glass is empty, what is keeping the water out?" When they say, "air," talk about the importance of being specific (full of what?), relating to the fractions and equations on the board (1/2 of what quantity?). You may also want to discuss when it is and is not appropriate to treat fractions as abstractions, or describe Empedocles experiment confirming air as a substance (a good account may be found in the book Who’s Afraid of Schroedinger’s Cat?).

Performance Assessment:

Group assessment should be made on basis of cooperation and success in involving and communicating with the rest of the class. You may also want to consider success in mathematically describing their experimental activities. Individual assessment should be based on participation in group and whole-class discussion and activities.

References:

Who’s Afraid of Schroedinger’s Cat? (paperback published in 1998; sorry, I don’t remember the author or publisher).