SECTION II:

Rainfall, Measurement, and Science

One of the phenomena that we are going to track throughout the year is rainfall. We often hear on the weather report that a certain amount of rain fell in a twenty-four hour period. We usually hear the total rainfall for the month reported as a number of inches. Our graph for identifying biomes reported total precipitation in centimeters. What does this mean? How do we measure rain? When a weather reporter states that an inch of rain fell, how is this being measured? Did the reporter measure the sizes of raindrops on their way down?

“Inches of rain” is an example of a phrase which is used more often than it is understood. Science is full of words and phrases like that. Atom, gene, momentum, compound, and precipitation are other examples. To really understand any of these things, we have to work up from the simplest levels, starting with things that we are absolutely sure we understand and working our way toward things that we only think we understand. It is only then that we can try to grasp the things that we know we don’t understand. This process is part of what is called “critical thinking” and it is a crucial part of scientific study.

The most difficult step in this process is often the first one since it requires sorting out the things we know from the things we only think we know (“prejudices” or “preconceived notions”). Throwing out our preconceived notions can be painful since it can leave us feeling as though we know less than we did when we started. In reality, we will be developing our intellectual integrity which will teach us about the boundaries of our own knowledge. This is one of the most important things we will ever learn.

EXERCISE: Think of an example from your life when you thought you knew something, but when you thought about that thing more carefully you realized that it didn’t make sense or just wasn’t true. Discuss your example with the students sitting around you. Discuss what led to these preconceived notions and more importantly what led to the realization that you were wrong. Record some of your observations and those of your classmates below.

This unit will be a careful trip down the path toward understanding how rainfall is measured. If we pay attention along the way, we can learn about critical thinking and the scientific method. Even if we don’t we will learn about measurement, area, volume, and rain.

MEASUREMENT:

Before we learn how to measure rainfall, we need to think about measurement itself.

  1. How many people are in your classroom right now. Take your time. Figure it out. Check your answer with your classmates. Record your answer as a complete sentence.
  2. How big is your classroom? Is this the same question as Question #1? What’s different about the two questions? Discuss the differences with classmates and record some here.

Answer #1. If we told you that you were going to be asked how many people are in another room and that the answer is seventeen, would you believe it? Is that even possible?

Answer #2. If we told you that you were going to be asked how big that other room is, and that the answer is seventeen, would you believe it? Is that possible? Why or why not?

Question #1 can be answered with a number. There can be four people in the room or there can be forty. You would never say that there are forty gallons of people in the room. Answer #1 is a perfectly reasonable answer. The answer could be seventeen.

Question #2 cannot be answered with only a number. Question #2 is asking about a quantity. We can’t get an idea of how big a quantity is unless we compare it to something else. For example, is your classroom a long room? Well, compared to a closet or a teacher’s office, it probably is. Compared to a bowling alley or a gymnasium it probably isn’t. To say that the room is “big” isn’t always helpful. We need to know how big it is, and for that we need to compare it to something else.

Sizes of objects are quantities and quantities must be measured. They can’t simply be counted. What do we need before we measure something? Let’s experiment by measuring a piece of paper.

MEASUREMENT EXERCISE: Find a marble. Your marble may not be the same size as your neighbor’s marble, but that’s okay. Don’t lose your marbles. Not yet, anyway.

1)Find a piece of paper. Oh my gosh! I am a piece of paper! Okay, you can use me.

a)How many marbles could you fit along the long edge of your piece of paper?

b)How many marbles could you fit along the short edge of your piece of paper?

c)How many marbles would fit on the surface (one side) of your piece of paper?

d)What is the relationship between the previous three answers?

2)Now forget the marble, but keep the piece of paper. Please.

a)What words might you use to describe the size of the long edge of your piece of paper? If you were to measure the long edge of your piece of paper, what units might you use for your measurement?

b)What words might you use to describe the size of the short edge of your piece of paper? If you were to measure the short edge of your piece of paper, what units might you use for your measurement?

c)What words might you use to describe the size of the surface of your piece of paper? If you were to measure the surface of your piece of paper, what units might you use for your measurement?

d)What is the relationship between the previous three answers?

3)Keep the same piece of paper. Please. I’ll be good.

a)How many gallons of water would fit along the long edge of your paper?

b)How many gallons of water would fit on the surface (one side) of your paper?

c)What is wrong with the previous two questions? What is a gallon anyway?

ACTIVITY: Estimate the area of the floor of the classroom. Now measure it. How can you measure it? (Hint: How big is one square of tile on the floor? What sort of units would you use to describe the size of a tile? How many tiles are there?)

So what would happen if we all measured paper (and everything else) using marbles? If you bought a piece of paper that was advertised to be 18 marbles by 13 marbles, could you be sure that 18 of your favorite marbles would fit along the long edge? What if the weather report predicted four marbles of rain? Is that a scary picture, or what?

What would we need to do before we started using “marbles” as units of measurement? Imagine that you are the emperor of the world and you are about to declare the marble as the new standard unit. You want everybody to measure everything in marbles. What would you have to do to make this idea work without forcing your empire into chaos? Think about all of the steps you and your people would have to go through to adopt your new system of measurement. Talk to your classmates to get feedback as to whether your ideas would work. Record your ideas and some of the ideas of your classmates here.

Once we all decide on a standard sized marble (or “inch” or “centimeter”) we need to make sure we understand the ideas of “area” and “volume” and how they are measured.

EXPLORING PERIMETER, AREA, AND VOLUME

Investigating rectangular areas with a fixed perimeter:

This is a project to be done in groups. You should work in groups of two or three people. Before you begin, take a few minutes to introduce yourself to the members of your group and discuss what you think is meant by the phrase “fixed” perimeter. When you think you have an idea of what is meant by an area with a fixed perimeter, record your idea here.

Now begin…

  1. Cut a piece of string 30 inches (in) long.
  2. With the piece of string, form a rectangle with a width of 1 in.
  3. Measure the length of the rectangle.
  4. Use the one square inch tiles to find the area of the rectangle. (Go ahead. Fill it up.)
  5. What are the width, the length, and the area of the rectangle? Make sure you use the correct units to answer each question.

WIDTH:LENGTH:AREA:

DATA TABLE:

Table 2-1:

Width
(in) / Length
(in) / Area
(in2)
  1. Record your answers in the top row of the table above. You will use this table to record and organize the same answers (“data”) for many rectangles.
  1. Now form a rectangle with a width of 2 inches. What is the length of the rectangle? What is the area of the rectangle? Record your answers in the table.
  2. Repeat this procedure to complete the table. Add more rows to your table if you need to, until you reach a largest possible area, and then do a couple more. (Do this enough times so that you see the area start to decrease a little bit.)
  3. Use a separate sheet of graph paper to make a line graph with the length and width pairs, using width as the input and length as the output. Be sure to label each axis with the quantity it represents as well as the appropriate units (gallons? inches?). How would you describe the shape of this graph?

  1. Copy your graph into the space provided below. Make sure to copy the labels and units.
  1. Using another sheet of graph paper, make another line graph with the width and area pairs, using width as the input and area as the output. How would you describe this graph? (Straight? Curvy? Wiggly?)
  2. Carefully copy this graph into the space provided below. Make sure to copy the labels and units. The numbers on the sides of the graph above gave you hints as to how to scale your graph. Here you have to supply the numbers yourself. Make sure that you leave yourself enough room to draw the whole graph. (Should one vertical grid mark be on unit, two units, or ten units? How many units should one horizontal grid mark be?)
  1. Are there widths and lengths other than whole numbers of inches that would give an area larger than the largest area you have recorded on your table? How do you know?
  2. Is there a relationship between the width and length of the rectangle and the length of the string that gives the largest area? How do you know? What assumptions are you making about the shape of your graph? How confident are you about those assumptions?
  3. Where on the width and area graph is the point representing the largest area? How do you know? Without actually filling another shape with tiles, can you predict the width that would produce the greatest area? Can you predict what the greatest area is?

Using data from a situation you have encountered to make a prediction or an inference about a situation that you have not yet encountered is the main purpose of scientific investigation. Without it nothing could be invented, explorers could not survive in deserts, in the ocean, or in outer space, and you would not know where to find the next set of instructions for this course.

Ah ha! You found them. You made an inference that they would be on the following page and you were right, so here they are: Pat yourself on the back.

THINKING ABOUT OUR EXPERIMENT:

Our experiment has accomplished a couple of things.

First, our goal was to learn how to measure perimeter and area how these two concepts are related. Different rectangles with the same perimeter can have different widths and areas, and by carefully organizing our data we found patterns in the widths, lengths, and areas.

Second, by predicting the width that would produce the greatest area for a rectangle as well as the area it would produce, we have extended our own measurements and data to a prediction of what would happen if we actually did the experiment in a different way. That required us to infer something about a new and different situation. This inference is an example of extrapolation: we have used the results of our experiment to make a prediction.

Are predictions important? Predictions are often associated with fortune-tellers, not with scientists, but experiments are useless without predictions. If we only learn what happened in the past when we did an experiment, we cannot apply our knowledge to the world around us. To be scientists, we have to carefully observe what can be seen and extrapolate our results to predictions of what would happen if experiments were repeated in different ways.

Are predictions reliable? That depends on the prediction. A scientific prediction is one that is backed up by such clear experimental results that we would be astounded if they were incorrect. Would you believe somebody who told you the maximum area of your rectangular string would actually come from a width of 3.5 inches? Of course not. We can reliably predict the width that produces the maximum area even though we didn’t actually see it.

ACTIVITY: Think of a prediction or inference you can make about something that will happen today or in the next few days. Think of a prediction that you can be very confident about. How can you be so sure this will happen? Compare your prediction with those of students around you.

DIMENSIONS

We often hear that the latest computer games come with new and improved “3D” or “three dimensional” graphics. Somebody who has only one talent and appears to be a bit of a dweeb is often called “one dimensional” (this would not be a good time to make fun of your instructor). What does “dimensional” mean?

If you want to describe the size of your piece of string, it is logical to describe it in inches or centimeters. We can say that the string is essentially one dimensional. To measure it we lay it flat on a table and hold a ruler next to it. We would usually state the length of the string, but we wouldn’t usually describe the thickness or the height above the table (unless there were many kinds of string available). One measurement is enough, so we say the string stretches out in one dimension.

The rectangles that we just laid out on the table and measured extend outward in two directions (which we called length and width). We say that such a shape is two dimensional and must be measured in area, not just length. The classroom floor is an example of a two-dimensional object. To describe its size, we needed to use words like “square feet” or “square meters.” We measured our rectangles in square inches. Measurements of two dimensional sizes refer to area.

When we buy milk, it doesn’t help if we tell the shopkeeper how long we want it. (“A long time, we’re going to drink it. Yuk, yuk.”) We can’t measure milk in square inches. We can measure milk in liters or gallons or even cubic centimeters (cc). How big is a liter? Find a ruler and draw a line ten centimeters long. Now draw a square that is ten centimeters on a side. That square has an area of 100 square centimeters (why?). Now imagine a cube which is ten centimeters by ten centimeters by ten centimeters (your teacher may have a few lying around). That cube has a volume of 1000 cubic centimeters which is the same as a volume of one liter. (In case you are wondering, a gallon is a little less than four liters.)

The same investigation that we did with the string and the tiles can now be extended to an investigation of volume, but a couple of things have to change. We covered areas with flat tiles. What will we need to replace our tiles in a study of volume? What did the string do for the areas covered by tiles? What do we need in place of the string if we are filling volumes?

What follows is an exercise on volumes and the surfaces that bound them.

INVESTIGATING VOLUME WITH A FIXED AREA:

  1. Cut, if necessary, a piece of paper to 20 centimeters (cm) by 28 cm. Not me! Put down those scissors! Get some paper from your instructor. Thank you.
  1. Cut a 2 cm square from each corner, as shown in the diagram.
  2. Fold the paper on the dotted lines to make a box without a top.
  3. Carefully tape the edges of the corners together.
  1. Measure the length, width, and height of the box. You will repeat this process with many different sized boxes. The length, width, height, and volume of each box, along with the size of the squares cut from the corners of the paper will be the data for this experiment.
  2. Fill in the table below by placing the data in the appropriate column. Using the unix cubes, calculate the volume of the box. (You might want to measure the unix cubes in order to have the volume measured in cubic centimeters (cm3 or cc) not in number of unix cubes.)

Table 2-2: