11. Job Rate Problems

The concept of “rates” provides many very popular word problems in algebra. “Betty can perform a certain job in 6 hours; Mary can perform the same job in 9 hours. Working together, how long would it take them to complete the job?” In truth, this is not only too philosophical, but the “answer” is impossible to verify. However, consider the following variation: “Hose A can fill a certain swimming pool in 6 hours. Hose B can fill the same pool in 9 hours. Working together, how long would it take both hoses to fill the pool?” This type of problem is not only very realistic and important, but we can also verify the answer. Here is what is required.

Find a nice container to act as the “swimming pool.” Typically, this will be a large “see through” right cylinder or rectangular prism. A certain height should then be marked to represent the level at which we declare the pool has been “filled.”

Now, our hoses are obtained as follows: Take a plastic gallon milk carton and completely remove (cut out) the bottom. Drill a hole in the carton’s screw-top lid. Finally, and this is most important as we will see below, completely surround the outside of the carton (a few inches from the removed bottom) with one strip of masking tape. To prepare various other “hoses” having different rates of flow, prepare other milk cartons in exactly the same manner, but use a different size hole for their lids. (Thus, you want a good variety of lids with holes made from different sized drill bits.)

Now, turn one of these bottles up-side-down (cover the lid’s hole with your finger), fill it with water, remove your finger and watch the water flow out. The thing you should observe is that the “rate” of flow is a function of the height of the water lever. That is, with a lot of water in the bottle, a greater volume of water will flow out of the bottle (per unit time) than when there is very little water remaining in the bottle. Therefore, each “hose” requires a little more to it, and here it is: While one student is operating a certain hose (i.e., a certain up-side-down plastic gallon milk carton), a teammate is simultaneously pouring water into its open top (the removed bottom), using another container, always being sure to keep the level of water (in our “hose”) within the width of the aforementioned masking tape! In this way, the rate of flow is kept very constant, and we can now model the swimming pool problem very nicely.

So, the experiment goes as follows: The students will time how long it takes each of two different “hoses” to “fill” the same “swimming pool.” (As in the Battery Operated Cars MathLab, you’ll need plenty of timers). Once that is known, the student will be asked, “If both hoses are now operated together (over the same empty “pool”) how long will it take to fill the pool?”

If they perform all aspects well, teams of students should be very successful with this MathLab. Thus, again, the class will have witnessed:

Q. A very practical question, “Knowing how long each hose takes to fill this pool, how long should it take both hoses, working together, to fill the pool?”

R. that is resolved using mathematics; and

V. the solution is verified by actually performing the experiment.

Collaborative School Activities for the Teaching and Learning of Mathematics