Teaching outline and Presentation suggestions for the materials on Pi, Pressure and Power.
(These suggestions are made without actually testing them with a real elementary school class. Look upon these as a point of departure in which other teachers will read and try our ideas and eventually, feedback from real classroom situations will cause these suggestions to grow into a useful body of knowledge. Let us know what worked for you and we will put it up for others to use.)
The meaning of pi:
Place the many different sized circular objects, each clearly marked with a unique identifying number (or letter), around the room at places where the students will be working. With students working in groups of four, supply each group with a paper metric tape (or string and a ruler). Each student can have a data sheet or each group could be responsible for one data sheet. Have the students record the identifying number for each object, give a brief description of the object, record the diameter and circumference of the object and then compute the ratio of the circumference to the diameter in a final column. (The teacher should decide if this shall be a math exercise in long division or if calculators can be used or--??) Keep passing the objects around until each group has measured several different objects. Don't expect to have all the ratios to come out the same. There will be experimental error and experience with high school students lead to a range of values for pi from about 2.9 to 3.3.
After the students have finished the activity and the results are in, it should be possible to have a discussion of how the measurements will vary but if this kind of experiment were repeated many times, and with great care, it would be discovered that all circles, large or small, will have the same value of the ratio of the circumference to the diameter. (It probably would not be appropriate to discuss that there are "non physical methods" of computing pi mathematically and that the numerical value can be specified to any desired degree of precision. It is of particular inters to mathematicians that this ratio is "not rational". That is, it can not be expressed as the ratio of two integers--22 over 7 is close, but the nature of pi prevents it from ever being expressed as the ratio of two integers.)
The following activities may help students (and others) finally to come to a real understanding of the meaning of pi:
1) Have students look up the history of pi. (There are several books devoted to this topic.)
2) Once they know the meaning and the approximate numerical value of pi, ask them: "What does pi mean?" Don't accept the numerical value (the usual quick answer) for what pi really means. Insist that they tell you that it is the ratio of the circumference to the diameter of a circle. Also insist that they appreciate that all circles, large or small, will give the same value for pi. (Naturally they don't have to use the exact words, even pictures are acceptable, however, make sure there answer shows that they have the meaning.)
3) Have students quiz other older students and even their parents about the meaning of pi. They might learn it better themselves as well as teach others its meaning.
The meaning of pressure:
In this activity it would be nice if each group of four students had a full set of all the nails and each student had a cardboard square. First the students should study the cardboard square and mark places on the cardboard where only a single layer of material stands over the corrugated inner structure. Since they will be punching through this single layer they will have to mark the surface of the cardboard in places where it is only a single layer thick. Next the group should discuss which of the nails will be the easiest to punch through the cardboard and which will be the hardest. Hopefully they will come to the conclusion that the smallest cross section will be the easiest and the largest cross section will be the hardest. After they have discussed the results, have each student punch his cardboard square with several different sized nails and see if the prediction was correct. (It won't always be clear which is easiest when the diameter differences are small but, there should be no argument between the largest and the smallest.) The students can take data on all of the nails by listing them by the numbers you have given the nails, perhaps they can use a sharp pencil and trace around the nail to show its cross section and then record some sort of subjective statement about how hard it was to push the nail through the first layer of cardboard.
When you discuss this after the activity be sure to remind the students that they were supplying a force (that is a "push") and a larger force was required to push the larger nail through the cardboard. Start discussing pressure as force divided by area. (We have found that students don't often know that dividing by a smaller number will produce a larger number. Perhaps a good way to help students to understand this is to discuss money. "How many half dollars are there in a dollar?" (2) "Now you know that a quarter is smaller than a half dollar. How many quarters are there in a dollar?" (4) "Now a dime is smaller still, how many dimes are there in a dollar?" (10). "Do you see as we divide the dollar by smaller and smaller amounts of money, it always takes more to make up the whole.") We hope that the concept of pressure will help you explain this mathematical fact as the students come to appreciate that a smaller area with a given force will produce a larger pressure.
Additional questions on pressure:
1) When you push a tack into the wall, why does the point go into the wall but the head of the tack doesn't go into your thumb? (Remind the students that the force is the same at either end of the tack--the difference is the pressure.) What would happen if you turned the tack around?
2) Explain why a sharp knife is easier to use when cutting a piece of wood than a dull knife. (When you sharpen a knife you are really making the cutting edge have a smaller area so you get more pressure for the same force.)
(The following question may be too difficult but we present it here for your consideration and we would like to know if it can be explained to young people. We suspect that explaining basic physics like this may require a higher level of maturity than we are assuming here.)
3) Why when the pressure of a tire is too low does it get flat on the bottom? (The point is that the low pressure tire must still hold up the same weight or force. The only way it can exert the same force when the pressure is too low is to make a greater area of contact with the ground.)
(Units of Pressure: Since in the United States we almost always measure pressure in pounds per square inch, we will use these units in the activities that follow. However you might be interested to know that the metric unit of pressure is the newton per square meter and it is given the name "Pascal". That is, Pascal=newton/square meter. This is a small unit compared to the pound per square inch. To give you a feeling for the relative size of these units, the pressure of the air around you is about 15 pounds per square inch or about 100,000 Pascal.)
The meaning of power:
It is assumed that you have done the "running up the stairs" activity in the unit on Force, Work and Power. If you have not, check out link to anchor on force work and power stairs running section. Most people are familiar with the watt as it relates to the power used by different electrical devices. (The watt is a unit of power and it makes no difference if you are talking about electrical power, mechanical power or nuclear power. Power is always work or energy per time and the watt is always a joule per second.)
In this activity we will put light bulbs of different wattage under the same quantity of water in an insulated cup and measure how much time it takes to heat the water some predetermined amount. A major problem with this activity is that the teacher should be in charge of doing the experiment because it does involve electricity in water. Activities which involve students watching and waiting, while the teacher manipulates the equipment, often causes difficulty with classroom control. We can imagine a situation in which all of the equipment will be set up a day in advance by the students. Each of several groups of students will be assigned to a different wattage bulb. It will be their responsibility to fill the cups with the same amount of water as the other groups, invert the bulb in the water and insert the thermometer in preparation for the experiment but the extension cord is not supplied so the experiment can't be initiated. In other words, small groups of students prepare the experiment, each with a different wattage bulb. On the day of the experiment (perhaps the following day), the equipment will be positioned where the teacher can start each experiment by plugging in the extension cord with special instructions that the students are only to read the thermometer and watch the time. If this method of running the activity is attempted, it is suggested that the smallest wattage bulb be started first and the largest last, so all groups will finish at about the same time. (Also, it might add to the excitement if sort of a "race" will be going on and the most powerful starts last.) In this way the students will only watch and take data with the teacher having full charge of touching the apparatus. A problem with this method is that the cup should be stirred occasionally. Would it be possible to assign a student to shake the cup gently or would it be better if the teacher returns to each group regularly to check what is going on and to stir the water? (Again we stress that this experiment is no more dangerous than using an immersion heater with the added safety feature that if the bulb is removed from the water during the experiment, there is no danger of burnout. However, there is always a small amount of danger when handling electrical equipment with water and it should be considered carefully.)
It will be agreed in advance that the water is to be heated some designated amount (perhaps 10 degrees C.) However, the students can take data on temperature and time which will keep them actively involved while the experiment is running. The only data needed, however, is the change in temperature of the water and the time required for it to change this amount of temperature. If it is agreed that the experiment ends when the water reaches a given temperature, and everyone starts at the same temperature, only the time needs to be measured.
(With this experiment it is possible to actually compute the amount of watts of heat the bulb "pumps" into the water. To do this you need to know the mass of the water in grams, the change in temperature in degrees C and the time required to make this temperature change. With the well established fact that 1 calorie equals 4.18 joules, and the fact that it takes one calorie to heat one gram of water one degree C, the actual wattage of the bulb can be tested. We mention this just in case you already know about it and you think your students might be able to appreciate it. However, it is our opinion that this kind of computation will be too difficult for young people to understand. At the same time it could make a special project for a very capable student. The equipment used in this experiment can be employed to test the wattage ratings of different bulbs.)
In the discussion following the experiment we hope the data will be good enough to establish that the higher wattage bulbs heat the water in a shorter time. Within experimental error, the data should show that twice the power will heat the water in half the time. We hope this will give you one more opportunity to stress that power is work or energy per time. If the same amount of water was heated through the same temperature change, the same amount of work (or energy) was done. More power just means that it can be done in less time.