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.
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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.
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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?
2)
3)
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?
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?
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?)
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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:
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DATA TABLE:
Table 2-1:
Width
(in)
Length
(in)
Area
(in2)
6. 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.
7. 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.
8. 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.)
9. 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?
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10. Copy your graph into the space provided below. Make sure to copy the labels and units.
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10
8
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4
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0
0
2
4
6
8
10
12 14 16
11. 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?)
12. 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?)
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13. 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?
14. 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?
15. 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.
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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.
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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 twodimensional 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.
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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.
2. Cut a 2 cm square from each corner, as shown in the diagram.
3. Fold the paper on the dotted lines to make a box without a top.
4. Carefully tape the edges of the corners together.
2 cm
2 cm
5. 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.
6. 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:
Corner
Length
Width
Height
Volume
Square
(cm)
(cm)
(cm)
(cm3)
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7. Repeat the calculations for a 4 cm square cut from each corner, then a 6 cm square, an 8
cm square, and so forth, until you can’t cut a bigger square from the paper. Record all of
your data in the table.
8. What is the largest volume of the boxes that you make in this way?
9. We often hear that the volume of a box is given by “length × width × height”. Why does
this mathematical formula give the correct volume? Look at your boxes full of unix
cubes. What is volume, anyway?
10. Using a separate sheet of graph paper, make a line graph with the corner square’s side
length as the input and the volume as the output. How would you describe this graph?
(Straight? Curvy? Wiggly?) Carefully copy this graph into the space provided below.
Make sure to copy the labels and units here as well. (How should you scale your axes?)
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11. Could there be a size of a corner square other than a whole number of even centimeters
that would give a volume larger than the largest area you have recorded on your table?
Try a few different values after looking at your graph.
12. Can you predict exactly the size of the square that would produce the greatest volume?
How confident are you in your prediction? Is there a difference between your confidence
in your prediction here and your confidence in the prediction of the maximum area of the
string rectangle? Why or why not?
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THINKING ABOUT VOLUME…
By now we should have improved our understanding of length, area, and volume, and the
relationship between them. Maybe you know a mathematical formula for the volume of a
box in terms of length, width, and height and you probably know how to use a mathematical
formula. We are often tempted to believe that the words “length, width, and height” tell us
everything there is to know about volume because that is all we have been told!
Even if you knew about length, width, and height, could you have predicted the shape of the
graph on the previous page? Could you have predicted the size of the corner square that
produced the most volume? The correct predictions are buried in “length, width, and height”
but they are buried so deeply that we would not find them without experiments (or advanced
mathematics). Thinking critically and investigating carefully have revealed much that we
didn’t know about something we may have thought that we already knew.
THINKING ABOUT PREDICTIONS…
Some experiments produce more definite predictions than others. How confident are you that
you know the exact size of a side to make with a 30 inch piece of string to make a rectangle
with the maximum possible area? How confident are you that you know the exact size of a
square to cut from a 20 cm by 28 cm sheet of paper to make a box with the maximum
volume? Both experiments produced predictions, but one produced greater accuracy.
Let’s think about the size of the “best possible” corner square, the square that we could cut
from that sheet to make a box with the greatest volume:
Could it be 6 cm on a side? Could it be 2 cm on a side? If you can answer these
questions, then you definitely learned something from your experiment.
Could the best possible square be 3 cm on a side? Could it be 5 cm on a side? Are
we getting close? What do you think the best size is?
Do you think the best possible size is 4 cm on a side? Exactly 4 cm on a side? Could
it be 3.8 cm on a side? Could it be 4.25 cm on a side? What would be the most
honest way to report your results for the size of the best possible square?
Somebody might say that the best possible square is 3.836668 cm on a side. Does it
make sense to say that your results show that a 3.836668 cm corner square will yield
the most volume? How would you make such a square? How would you measure it?
Sometimes we are tempted to make statements that appear to be very accurate, even though
we don’t really know our answers that accurately. This is not a good thing to do since it will
make others believe that you do know your answer that accurately and they may expect you
to be able to prove it! In general we should remember to report our results with only as much
accuracy as we really know. Scientists sometimes refer to this process of being honest about
their accuracy as keeping track of significant figures. We will see more about that later.
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THINKING ABOUT RAINFALL…
Remember rainfall? Yes, we’re still talking about rainfall here. Rain is water. If you need to
know how much water a fish tank will hold, what units (and dimensions) do you need to
know about? A tank can’t hold “forty water” or “forty minutes of water.” If you want to
know how much water an aquarium will hold you need an answer in terms of volume.
We measure volume when measuring quantities of water. So why do we talk about “inches
of rain?” Weather stations collect the rain in containers called rain gauges. By measuring
the water in the rain gauge, they determine the amount of rainfall. (Sorry, a rain gauge that
has an inch of water in the bottom does not necessarily mean we had an inch of rain. It is
more complicated than that, but don’t worry. We’ll get there.)
Rain gauges come in all sorts of shapes and sizes. Some are tall and narrow. Some are short
and wide. Some are wide at the top and narrow at the bottom. Notice that the boxes you
made out or paper also had very different shapes as well as different volumes. Is it possible
for boxes to have different shapes yet still have the same volume?
Investigating different shapes with a fixed volume:
1. Using 36 unix cubes, create as many distinct boxes as possible. You will need to
determine the length, width and height of each box so that the total volume is 36 cubes.
2. Fill in the table below with as many possibilities as your group can find. Once again note
that each unix cube is 2 cm × 2 cm × 2 cm with a volume of 8 cm3. (So if you use 2 unix
cubes, what is the volume in cm3 ? What if you use 3 cubes? How about 36 cubes?)
Table 2-3:
Length
(cm)
Width
(cm)
Height
(cm)
Volume
(cm3)
3. Make several of these boxes out of the cubes. Share your results with other groups. Why
might someone use a box with each of these very differently shapes? (Think about some
practical uses for these shapes…)
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Now comes the fun (wet) stuff! We get to make it rain!
MEASURING RAINFALL, PART I: USING RECTANGULAR CONTAINERS
Returning to our original question, what does it mean to say that we just had “one inch” of
rain other than that we are really wet? Through the last few activities we learned that inches
measure length or one-dimensional objects. But common sense tells us that rain needs to be
measured in containers which means using three-dimensional objects or volume. Do you
remember what units are used to measure volume? They were not “inches”. Maybe we are
smarter than those folks who tell us the rainfall! (We knew that anyway.)
MAKE A PREDICTION: Imagine that we leave two containers outside during a rainstorm.
One of the containers is wide and flat but not very tall (a cake pan or a baking dish would be
a good example). The other container is tall and very thin like a tube with the bottom end
sealed off. A picture of the two containers is shown below.
Flat container
Tall container
We leave the two containers out for the same amount of time. A lot of rain falls but not
enough for either container to overflow. When we collect the containers again, how much
water is in each container? How does the water in the two containers compare?
Make a prediction and discuss it with your classmates! We are going to try this out. Record
your prediction below… (Remember, predictions are not graded! They are a way to get us to
check our assumptions and start thinking about a problem. The most important predictions
in science are always the ones that turn out to be wrong!)
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ACTIVITY: Now we will try it out. Collect several open rectangular containers with
different shapes and sizes. Take a good look at your containers. If you placed these outside
on a rainy day in the same location for the same length of time, what would you expect to
happen? Would the amount of rain in each container be the same? When we talk about
amount of rain, to what dimension(s) are we referring? Would the height of the water in each
container be the same? When we talk about height, to what dimension(s) are we referring?
In your group, discuss again what you think will happen. Write your prediction!
1)
Measure the base area of each of your containers. How are you going to do this? What
shape is the base of the container? How did we measure these areas before? What are
the units for the area? Record your results. (We will record them in a table later.)
2)
Imagine that you have one rectangular box with a base that has an area of twelve square
centimeters. A picture of the base is sketched below…
If this box had a height of 1 cm, how many 1 cm3 unix cubes could you fit in the box?
What would the volume of the box be? Are the answers to these questions the same?
3)
If that box had a height of 2.0 cm, how many 1 cm unix cubes could you fit in the box?
What would the volume of the box be? Are the answers to these two questions the same?
4)
If that box had a height of 2.5 cm, how many 1 cm unix cubes could you fit inside the
box? What would the volume of the box be? Are the answers to these questions the
same? What would you have to do (to the cubes) to make the answers the same?
5)
Pretend that you have to explain your answers to the questions above, to a child who
does not know how to multiply. What would you say? What words would you use?
The concept you are trying to relate is called proportional reasoning and it is one of the
most useful tools in all of science. Discuss with your classmates how you might explain
this proportional reasoning to a child.
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6)
In the room you will find several “rain machines”. A rain machine is a box that would
be water tight if it weren’t for all of the holes somebody punched in the bottom. Using
your rain machine, make it rain over your collection of containers for a specified amount
of time. (You may find you have enough water after ten seconds or you may need 30
seconds, but once you decide how much time you need, make sure that each container is
rained on for the same length of time!)
Measure the volume of rain in each container. Record your results in the following table.
DATA TABLE: fill in the FIRST FIVE columns now
Table 2-4:
Length
(cm)
Width
(cm)
Height
(cm)
Area
(cm2)
Volume
(cm3)
Volume/Area
7)
What do you notice about the quantities entered in the “Height” column for each of the
containers? What do you notice about the quantities entered in the “Volume” column for
each of the containers?
8)
The sixth column asks you to divide the volume by the area. On your calculator this
means to type in “volume ÷ area =” . Think about what the units for this number might
be. What does it measure?
9)
Using a separate sheet of graph paper, make a line graph with the base area of the
containers as the input and the volume as the output. How would you describe this
graph? (Straight? Curvy? Wiggly?)
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10) Carefully copy your graph into the space provided below. Make sure to copy the labels
and units here as well.
0
This graph (and your data table) demonstrate an important relationship between area and
volume of a container during rainfall. This relationship is called “direct proportion.” The
graph above is a picture of the relationship that you described with proportional reasoning.
One way to think about a direct proportion is to ask, if the input variable changes does the
output variable increase or decrease? If the answer is that the output always increases with
the input, we suspect that there may be a direct proportion between the two quantities. If the
output increases by the same amount every time that the input increases by a certain amount
(and the output is zero when the input is zero), then we have a direct proportion.
On this graph, think about an ant moving from one little grid mark along the input axis (the
horizontal axis which represents area in this case) to the next little grid mark as a “step” to
the right. You can imagine starting out at the “0” in the bottom left corner and then taking a
step to the next grid mark to the right. How much did the area increase between these two
marks? Looking along your graph, how much did the volume increase?
Now take a step starting from that grid mark to the right of the “0” to the next grid mark to
the right. How much did the area increase between those two grid marks? Look along the
graph again. How much did the volume increase?
Your graph should show that the volume increases by the same amount every time we
increase the area one “step” to the right (as long as our steps are all the same size). This
graph is a picture of direct proportion.
Think of a relationship between two things that may have a direct proportion. (Use your own
experiences.) In your group, come up with at least three different relationships that you think
may be examples of direct proportion. List some of the relationships here. Discuss why you
think these are direct proportions.
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We can express the direct proportion between volume and area using symbols.
volume = (some quantity) ∗ (area) ,
(where " ∗ " means to multiply) .
This is called an equation. All that it says is that if the area increases, so does the volume, or
that volume is the same as some fixed quantity times area. You may have already figured out
what the ‘some quantity’ is in this case. However let’s pretend that we don’t know what this
number is (or what it represents). Okay, we do know that
volume = height ∗ length ∗ width .
We also know that “ length ∗ width ” is the same as area. So let’s substitute the word ‘area’
for “ length ∗ width ”. We end up with the equation
volume = (height) ∗ (area) .
Compare this with our first relationship:
volume = (some quantity) ∗ (area) .
We now have a strong suspicion that our ‘some quantity’ is really the height of the rain in the
container! Note that if we divide both sides of the equation “volume = (some quantity)*area”
by “area” we get
volume (some quantity) ∗ area
=
area
area
or
volume
= (some quantity) .
area
Since we now suspect that our ‘some quantity’ is really the height, we suspect that
volume
= height .
area
What we have done so far is called a derivation. We have strong logical reasons to believe
that our ‘some quantity’ is exactly the same thing as ‘height’. We could stop here and be
smug about how smart we are, but part of critical thinking is making sure that you really
know the things you think you know. This may seem obvious, but it really isn’t.
Think about the word “height.” What is your height? Take a moment to explain to the
person sitting next to you what you mean when you use the word height. Listen to what your
neighbor means by that word. Pretty similar? Ask each other how you would figure out what
the height of something is. Write down what you think height means.
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On the last page you should have written down some sort of definition of the word “height.”
Here are a couple of definitions:
1. Height: how tall something is.
2. Height: hold a ruler or a meter stick or some other device for measuring length up
against a wall. Make sure your ruler (or other device) is arranged so that the end
with a “0” on it is on the bottom (so it measures distance from the bottom to the
top). Stand an object up next to the measuring device and read the length that is
level with the top of the object. That length is the height of the object.
The first definition is entirely accurate but not always very useful. If you don’t know how to
find the height of an object, you probably don’t know how to find out how tall it is.
The second definition is longer, but it tells us what we need to know. The second definition
assumes we know how to measure length, but even if we didn’t, the definition informs us that
measuring length is something that we need to do to understand height. This definition tells
us what we have to do to know what height is. It is an operational definition. We will
discuss operational definitions more later since they are central parts of science.
Talking to your classmates should have convinced you that everybody in the room has some
idea of what “height” is. We each have an idea about height that we have grown up with
since childhood. Your idea of height is your own personal operational definition of the word.
Now look back at the derivation on the last page. Here we have a new discovery about
height. Our discovery is that for our little rectangular containers
volume
= height
area
This says, “height is the same thing as volume divided by area.” Using logical reasoning we
carried out a derivation that shows us why this should be true. Still, this tells us something
about “height” that does not sound the same as our usual definition. Does this new idea
about height agree with our old idea? Could it be that when we use the word “height” this
way that we are talking about something different than what we were talking about before?
There may be a skeptical little voice in the back of your mind asking whether our new
discovery is really true. Skeptical little voices are good things! They make us check our
assumptions! They make us think critically! We need to make sure that this new quantity
“volume/area” is the same as what we usually mean by “height.”
Look back at your table and compare the columns for height and the ratio of volume/area.
These two columns should be almost exactly the same. Why would there be any differences
in values? (Think about how you measured the container.) Record your observations.
19
THINKING ABOUT RAINFALL…
So what do we know at this point about measuring rainfall.
1)
If different sizes of containers are left out in the rain for the same amount of time,
what is true about the height of water in each container? ? Do containers with
different areas collect the same height of a column of water?
2)
If different sizes of containers are left out in the rain for the same amount of time,
what is true about the volume of water in each container? Do containers with
different areas collect the same volume of water?
3)
If the heights of the columns of water in the containers are the same, what is the
relationship between volume of rain in each container and the area of each container?
4)
Imagine a student in your class turns to you and says, “If you put a container outside
in a rainstorm, the volume of water that falls into the container will be in direct
proportion with the area of the container.”
5)
a)
Based on what we have done so far, can you think of any evidence to convince
you that this is true. What have you seen to make you believe this?
b)
Now can you think of an argument as to why this should be true? This is not
the same question as part a. We are not asking for the result of an experiment
here. Try to think of a logical reason why the volume of water that falls into
a container in the rain will be in direct proportion to the area of the container.
Since different rainfall gauges have different volumes and areas, height is the one
consistent measurement for all gauges. Since height is a one-dimensional concept,
what are the most reasonable units for measuring rainfall?
20
THINKING ABOUT DEFINITIONS…
If we all saw a duck swimming on a lake, most of us would be able to agree that yes, that’s a
duck. Somewhere in the backs of our minds we have an operational definition of the word
“duck.” For most of us, that definition reads something like, “If it looks like a duck, and it
walks like a duck, and it swims like a duck, and it quacks like a duck, then it’s a duck.” This
is sometimes referred to as the “duck test.” It is an example of an operational definition.
NOTICE: The operational definition of a duck can’t be just a new word or a concept. The
dictionary says that a duck is “a member of the family Anatidae” but that definition doesn’t
help many of us decide whether or not some bird is a duck. That definition is fine, but it isn’t
an operational definition (unless somebody gives us an operational definition of Anatidae).
An operational definition must tell us what to do to decide whether or not some thing (or
concept) is what we think it is. Operational definitions have some activity (and judgment)
associated with them. Our duck definition requires us to look at the duck as it swims and
walks. We could just decide to call something else by the name “duck,” but that wouldn’t be
science. An operational definition includes an active test of “duck-ness.” Science depends
on our ability to make observations and agree about what we observe. Scientists can’t define
anything without operational definitions.
It is always possible that after becoming convinced that we are looking at a duck, a wildlife
biologist will appear from the bushes to tell us we are looking at a loon or a coot or a mutant
chicken. Our operational definition might not be perfect. The biologist may have a better
operational definition than ours. Every day we use operational definitions for all kinds of
things. Most of them are useful, but any or all of them may be incorrect or incomplete.
Quick! Without consulting your classmates, think about your own personal operational
definition of the concept of “volume” and write it down. (Be detailed enough that someone
else could read it and understand what you mean.)
Now compare… compare notes with your classmates. Did somebody write down something
that has to do with the number of cubes that can fit inside? Did somebody write down
something that has to do with length, width, and height? Did somebody write something
else? Make a few notes to remind you of each of the definitions you heard.
21
Now check… critical thinkers always check their ideas. Do all of the definitions work?
Imagine that somebody asked you to find the volume of a soda can, a can of tennis balls, or a
tennis ball itself.
How neatly will unix cubes fit inside of a pop can? Is this a good way to measure the
volume of something with that shape? Why or why not?
Think about a tennis ball. The length, width, and height are all the same, right? What
will you get if you multiply the length times the width times the height? Is this the
volume of the ball? If not, what is it?
Will any of the operational definitions of volume that you have heard correctly
describe the volume of a round object? (Remember, an operational definition of
volume must tell you what to do to find the volume.)
Up to now, all of the containers that we have used to measure rainfall have had square
corners and straight lengths and widths. Most real rain gauges are round, but if we
investigate circular shapes, our square tiles and cubic-shaped unix cubes won’t fit!
In the room are examples of actual rainfall gauges. Notice that these gauges are not
rectangular in shape. We need to build some understanding of length, area and volume of
circular objects before we can understand how the containers actually work. Our goal will be
to develop a quantitative understanding. This means that we want to be able to understand
exactly what the area and volume of a round object is and why we get that exact answer.
Right now, before we work on quantitative understanding, think about your qualitative
understanding. That is, if we are trying to find the area of a round object, do you even know
what we are looking for?
Try one more time… Try to write down an operational definition of area (this is a little
easier than volume) that explains what you mean by the words “area of a circle.” Remember,
your definition has to be operational. Compare definitions with those of your classmates.
Also, before we go on, you might also notice that some of the rain gauges are not as simple as
our single-shaped containers. Some are bigger at the top than they are at the bottom (some
are “funnel-shaped”). That is another complication. We will get to this soon, but for now
let’s think about the circular shape. Take a look at what the Greeks discovered millennia ago.
22
CIRCLES AND SOME OLD GEOMETRY: ROUND THINGS AND LENGTHS…
1)
Using a compass (or some round objects) and a piece of graph paper, draw several
circles with different radii. If you use a compass, the point of the compass will be at the
center of the circle. Make sure you label the location of the center. If you use round
objects you will have to be a little more clever about locating the center, but you can do
it if you are careful about the placement of your circles at the start. Try to draw a few
circles with centers on intersections of squares on the graph paper. If you are using a
compass, make the distance between the point and the pencil lead in the compass a
whole number of centimeters.
2)
The distance across a circle is called a diameter. Half of the diameter is the radius of the
circle, so the radius is the distance from the center to an edge (if you used a compass, the
radius of each circle should be a whole number of centimeters). The circumference of
the circle is the distance around the outside of the circle. Since you can not lay a straight
ruler along the edge of the circle, try wrapping a string around the circle and measuring
the length of the string. You can also measure the edge with a bendable ruler or tape if
you have one. Record the radius, diameter, and circumference of each circle in the table
below. You can round your measurements to the nearest tenth of a centimeter (which is
also called a millimeter).
Table 2-5:
Radius (cm)
Diameter (cm)
Circumference (cm)
3)
Once you have your data, compare the measurements in the diameter column with the
measurements in the radius column. Are they related? (Did you measure each one
independently?) How are they related? What kind of relationship is this?
4)
Now compare the measurements in the circumference column with the measurements in
the diameter column. Do they appear to be related? How can you tell? What kind of a
relationship do you suspect they might have? Why?
5)
We might guess at this point that we are looking at an example of direct proportion.
How did we recognize direct proportion before? What did we do to see a direct
proportion?
23
6)
On separate graph paper, make a graph of your diameter and of circumference data. Use
diameter (D) as input data and circumference (C) as output data. Think of each diameter
and circumference pair as a “data point” on your graph, and show all of your points on
the same graph. Mark each point with a dark dot or very small circle.
By the way, you probably know another diameter and circumference pair that you did not
write into your table. If a circle had a diameter of “0 cm” what would the circumference
be? Most of the points on your graph are the results of pairs of measurements and
measurements have uncertainty in them. How much uncertainty is there in this new data
point? This new point is important. Include it on your graph.
7)
Now look at all of the points on your graph. If you were to “connect the dots”, what
would the shape of the graph be (straight, curvy, wiggly)? Your points should lie along
a fairly straight line. The line should go through the new point “0 cm, 0 cm” (we call
this point “the origin”). Try laying a ruler along your data points. If one of the points is
far from the ruler, you probably made a mistake with that one so check it again. When
all of your data points lie pretty close to a straight line that goes through the origin, use a
ruler to draw the line that seems to come closest to all of them.
8)
How many points did you plot on your graph? Five? Ten? Based on what you see,
describe what you think the graph would look like if you plotted a hundred points.
Imagine you used circles with radii of 0.1 cm, 0.2 cm, 0.3 cm, 0.4 cm… all the way up to
9.9 cm and 10.0 cm. What would the graph look like? How do you know?
9)
You can use your reasoning from part 8 to answer questions about circles that you did
not actually draw.
a)
Look at your graph. Can you find the place on your graph where a circle with
a circumference of 9 cm would be? Use your graph to find out shat diameter it
would have. (This is another example of using extrapolation to make a
prediction or inference.) What would the diameter be?
b)
Look at your graph again. Find the place where a circle with a diameter of
0.5 cm would be. What circumference would it have?
c)
Look at your graph again. Find the place where a circle with a diameter of
2.5 cm would be. Use your graph to find out what circumference it would
have. (This could be called another example of extrapolation, but since you
probably have data points on both sides of 2.5 cm, this would usually be called
interpolation. If you know what will happen when “you have more” and when
“you have less,” interpolation is used to make predictions or inferences about
what happens in between.) What would it be?
24
10) Your inferences about the 0.5 cm and the 2.5 cm circles probably came from the “bestfit” straight line that you drew in part 7. As such, they tell us something about the
straight line that you believe accurately predicts the behavior of your data. Record the
circumferences on the lines below so that we can learn about this straight line.
Table 2-6:
Diameter:
Circumference:
0.5 cm
______________
2.5 cm
______________
Change in
Diameter:
Change in
Circumference:
“Change in circumference for
each change in diameter”
11) If we want to tell somebody about that line, we might say it is straight and it goes
through (0 cm, 0 cm). Then we have to tell them how steep it is. To describe how steep
something is, we must describe how much it “goes up” with each step we take.
a)
On the line on our graph, we take steps to the right. A step to the right shows
a change in the diameter of a circle since “diameter” is on the axis that goes
from left to right. Imagine a 0.5 cm circle got bigger and increased to a 2.5 cm
diameter. Write the change in diameter between our two circles in our table.
b)
Now describe how much the line “goes up” when we take a step from a
diameter of 0.5 cm to 2.5 cm. On our graph, “up” is the “circumference
direction” so we need to know how much the circumference increased. Write
the increase in circumference in the table.
c)
We are choosing to use the centimeter as our unit of measure here. In the
table above, the diameter increased by how many units?
d)
By how many units would the circumference have increased if the diameter
had increased by only one unit? (Proportional thinking strikes again!)
e)
Look along the horizontal axis (the input or diameter axis) of your graph.
Choose some place (any place) on the graph and move one unit to the right.
How much did the circumference increase? Compare your result to your
answer in part d. Is the answer the same for every place on the graph?
f)
The quantity in part d tells us the “steepness” of the line, so we call it the
“slope.” In this case the slope tells us something about circles. In words,
what does this quantity tell us?
25
12) Explain in words how you would find the quantity in the last column in Table 2-6 on the
previous page (“Change in circumference for each change in diameter”) if the change in
diameter had not been two centimeters? What would you have done if the change in
diameter was 3 cm? What if it was 5 inches?
13) Mathematically we can write the slope of our line as
(Change in Circumference ) .
(Change in Diameter )
If Juan measured the circumference and the diameter in centimeters, and Mary measured
them in inches, how would these answers compare? (Would one be bigger than the
other?) Talk to your classmates, make a prediction, and record it with your reasoning.
14) Okay, now try it. You have already done the experiment as “Juan” so see what happens
if you do it Mary’s way. 0.5 cm is the same as 0.2 inches. And 2.5 cm is the same as 1.0
inches. What do you get for the slope if you do everything in inches? How does it
compare to the slope when you did everything in centimeters?
15) If we wanted to record a number that tells us the average height of the people in the
room, that number would depend on the units that we use. Why? If we want to record
the number of people in the room, that number does not depend on the units that we use.
If a certain quantity seems to be the same no matter what units you use, then does that
quantity have units? Discuss why this may be the case for the slope of our line.
26
CIRCLES AND SOME OLD GEOMETRY II: ROUND THINGS AND RATIOS…
Table 2-7:
Radius
(cm)
1)
Diameter
(cm)
Circumference
(cm)
Circumference/
Diameter
Yes, it’s the same old table again, except we’ve added an extra column. We now have
strong reason to suspect that there is a direct proportion at work here. We have reason to
believe that the circumference and diameter are related by
(Circumference)=(some quantity) ∗ (Diameter) .
We are now equipped to find out what that “some quantity” really is. Fill out the table
and this time fill in the fourth column as well.
2)
We forgot to tell you the units for the fourth column of the table. To makes things
simpler think about a square instead of a circle. If you did an experiment with a square,
and you divided the “circumference” by the length of a side, you would get “4.” What
does the number 4 tell you about a square? Are there units that go with that number?
What happens to the units in your measurements of the square? What happens to the
units in your measurements of the circle when you fill in the last column of this table?
3)
Why do we get this result for our units here? Think about what division means. Imagine
that we wrapped a string around a soup can and then cut it so that it had the same length
as the circumference. We could then lay the string straight and measure the length of the
string in “soup can diameters.” How many diameters would it be? (Try it if you want!)
Can you explain in words what the quantity in the fourth column means? Can you
explain why the units are the way that they are in each column of the table?
4)
What observation can you make about the numbers in the last column? Look at all of the
numbers in that column. How different are they? How accurately do you know each of
the numbers? Can you be sure that the first row in the last column is different from the
second? From the third? Could they be the same number?
27
Every measurement includes some error. You may have thought one of your circles had
a radius of 3.00 cm, but it may have been 3.01 cm. We can’t be sure. Since we can’t be
sure about the radii, we can’t be completely sure about the numbers in the last column.
Maybe all of those numbers are really measurements of the same thing. As a new
hypothesis, suppose the numbers in that column really are slightly different estimations
of the same number. Call the number what you want. “Fred” will do.
5)
Are any of your measurements of “Fred” really different from the others? If so, maybe
you made a mistake with that one. (Was that “data point” close to the line on your
graph?) Are any of them so far from the others (notice that we’re not asking if the
number is far from what you want it to be) that you are convinced it must have been a
mistake? If so, you may want to redo or ignore that one.
6)
To get the best estimate of how big “Fred” really is, we can assume that each of our
measurements is close, but has some error in it. Imagine you are trying to cut 5 cm strips
of paper. Some will be too long and some will be too short. You might guess that all of
the strips of paper will be 5 cm long “on the average.”
If we find the average of all of our estimates, it will give us an better estimation of Fred.
Find the average by adding up all of the numbers in the last column and dividing by the
number of numbers. If you have five numbers, add them up and find the sum. Then
divide the sum by five. That number is your average. Don’t write down all the digits on
your calculator. Since you rounded off your measurements, you should round off your
average, too. Look at the first three digits (this is what scientists would call “three
significant figures”). What number do you have? Do you recognize this number?
Discuss your observations about this number with your classmates. See if you can come
up with a name that might be more useful than “Fred.”
7)
How does this quantity compare with the number you got for the slope of the line on
your graph? What might be the relationship between this number and the slope?
Discuss this with your classmates, formulate an answer, and explain your reasoning.
8)
What would you get if you divided the circumference of a circle by the radius? Try to
write an equation of the form Circumference = (some quantity) ∗ (radius) .
28
ROUND THINGS AND AREA…
1)
Measure the area of several circles drawn on graph paper by counting squares and
estimating the partial areas. Make your own table similar to the one we used for radii
and circumference to compare the values of the radii and the approximate areas. If you
need a fourth column for this table there is space for an extra one below.
Table 2-8:
Radius
Diameter
Area
_________
To get a more accurate measurement of area we need to discover some kind of a
formula to help us. The first tool that we used to study area was the rectangle, so we
will return there now. (Critical thinking: use only the things that you are sure you
know to learn about the things that you want to know.) We are going to try to turn a
bunch of circles into a bunch of rectangles. Then we can figure out the area.
Divide into groups, and have each member of your group get a paper circle with
diameters drawn across it from your instructor.
2)
3)
Before we begin, the lines drawn across the circle (“diameters”) divide the circle into
pie-shaped pieces called sectors.
a)
Use a protractor to measure the size of each sector in degrees (measure how
many degrees there are in the angle the formed by the point of the sectors).
b)
How many degrees are there in your whole circle?
Cut along each diameter line in your circle. You should have a set of sectors of the
circle. How long are the straight edges of each sector? What does this tell us about the
original circle?
29
4)
Let the length of the straight edge of each sector be represented by r . (Why do we use
that letter?)
5)
What is the circumference of your circle? Do you have to measure it or can you figure it
out from something you learned in the last exercise?
a)
Write down the length of the circumference as a quantity (like 12 cm).
b)
Now write it down as a mathematical expression in terms of r (such as 6r).
6)
Let the circumference be represented by C. (Why? oh good, you’re catching on.)
7)
Arrange the sectors from only one circle on the table to make a shape like this:
8)
Compare the shape you get with the other members of your group. Some members of
the group have a lot of sectors, others have just a few. If you look at the shapes with just
a few sectors and compare them to the shapes with lots of sectors, what do you notice?
What shape do the circles cut into lots of sectors seem to form? As the number of
sectors in each circle increases what happens to the shape?
9)
Imagine that you could cut out 360 sectors (each sector one degree in width). The shape
you would get is very close to what geometric shape?
10) Your instructor should have some rectangular sheets of paper. (If you are lucky the
paper is even sticky.) Take the circle from your group with the most sectors and lay the
sectors over the rectangle. What can you say about their areas?
30
11) Compare the circumference of your original circle to the length of the rectangle. How
does the circumference of the circle compare to the length of the rectangle? Why?
12) If r is the radius of your circle, what is the length of the rectangle? Why?
13) Compare the radius of the circle to the width of the rectangle. If the radius of your circle
is r, what is the width of the rectangle? Why?
14) How do you find the area of a rectangle? You now know the length and width of this
rectangle in terms of r. What is the area of this rectangle? What does that tell you about
the area of the original circle? What is the area of the original circle?
15) Imagine somebody gives you a round object, a circle. Explain how you would find the
radius of the circle. Explain how you would calculate the area of the circle.
16) Now go back to the table you made by estimating the area of the circles. Use what you
now know about areas of circles to try and calculate the area of each circle from the
radius. Compare the calculated areas to the areas you measured by counting squares.
Fill out the table below to get an idea of how well the two methods for finding area
agree. The table also includes a column for “error.” How would you fill out that
column? Which method for finding the area do you think is more accurate? Compare
your data and your ideas about accuracy with those of your classmates.
Table 2-9:
Radius of circle
Measured Area
31
Calculated Area
Error
BACK TO RAIN… AND PANCAKES:
When we measured rainfall using our rain machines, we found that the amount of
water in any container depended on the container itself, but that the “height” of the
column of water in each container was the same. In order to understand rainfall we
had to learn about both area and volume and we had to learn the relationship between
them. This is what led us down the path of measurements, proportional reasoning,
and predictions. Still, all of our rain collectors had rectangular bases.
For a round object, a cylinder, imagine that the base of the cylinder is a circular
pancake (a tortilla, a chapati, or whatever). Now imagine that the whole cylinder is
just a stack of pancakes. How hungry are you? You could say that you want three
pancakes, but that really depends on the size of the pancakes. The amount of food in
your stack of pancakes is just the number of pancakes (the height of the stack) times
the size of the pancakes (the area of the cylinder).
Before we jump into another mathematical formula, let’s try out the proportional
reasoning skills again. Imagine that the circle below is the base of a cylinder and that
the circle has an area of twelve square centimeters (often written as 12 cm2).
1)
Explain in words what it means to say that the circle has an area of 12 cm2.
2)
If the cylinder were only one centimeter tall, what would the volume be? Explain in
words how you know.
3)
If the cylinder were 2.5 cm tall, what would the volume be? Explain in words how you
know.
32
Now imagine that the star below is the base of a star-shaped container. The container
itself is shown on the right.
Side view
4)
If the area of the star is 9 cm2 and the height of the container is one centimeter, what is
the volume of the container?
5)
If the area of the star is 9 cm2 and the height of the container is 5 cm, what is the volume
of the container?
6)
Can you write down a mathematical formula for the volume of any container with
straight sides? What two pieces of information do you need to find the volume?
7)
Can you write down a mathematical formula for the volume of a circular container with
straight sides (a cylinder)? Can you write down a formula that works if you only know
the radius and the height?
33
By now you should be able to understand the formula for the volume of a cylinder:
Volume = π r 2 h
where r is the radius of the base and h is the height of the cylinder. What part of this
formula tells you about the direct proportion that you used on the previous page?
BACK TO THE RAIN MACHINES!!!
Now that we have an amazing understanding of volume, joyfully rush back to your
favorite rain machine and make it rain over a bunch of cylindrical containers (soup
cans, tuna fish cans, toucans…). Make sure that it rains for the same amount of time
over each container. What do you notice about the water in each container?
1)
Is the amount of water the same in each container?
2)
Is the height of the water the same in each container?
3)
The volume of a cylinder of water is directly proportional to what two properties of the
cylinder? (HINT: The volume is not directly proportional to r itself since r is squared in
the formula at the top of the page. Volume is directly proportional to _______ and
_________.)
4)
Imagine that it is raining on a soccer field. Pretend that you know fifty gallons of rain
will fall on that soccer field (soccer fields are big, so only a little falls in any one spot!).
Now imagine that we set a container in the middle of the field. The container could be
an inflatable swimming pool, it could be an aquarium, or it could be a coffee cup. What
property of the container determines how much of the fifty gallons falls into the
container? Explain your reasoning.
5)
In any rainstorm, the amount of water that falls into each container must be directly
proportional to what property of that container? Can you draw a picture of rain falling
into and around the container to show why this is true?
34
We are about to assemble the pieces, so we need to do a quick check to make sure we
have them all…
6)
7)
If you set a container outside, the amount of rain that actually falls into the container
depends on a property of the container and something else. Those two things are:
a)
The property of the container called ______________________
b)
Something else which is _______________________________
When you bring the container back in it has a column of water in it (assuming that it
actually rained!). The volume of water in that column is directly proportional to:
a)
The property of the container called ______________________
b)
A property of the column of water called __________________________
8)
Think of the property of the container that you identified in parts 5 and 6 (above). If the
amount of water that falls into each container is proportional to that property of the
container, then why isn’t the height of the column of water in the container also
proportional to that property? (HINT: look at your answer to part 7)
9)
We have not yet come up with a formal definition of the idea of “rainfall” except that it
tells us “how much it rained.” That is what we are trying to understand here. One of
your classmates might say that the volume of rain that accumulates in a rain gauge is
proportional to the area of the top of the rain gauge times “how much it rained:”
( rain in rain gauge ) = (area of rain gauge) ∗ ("how much it rained") .
Could this be used as an operational definition of rainfall? Explain your reasoning.
10) We know something else about the volume of the rain in a rain gauge:
of column of water)
( rain in rain gauge ) = (area of rain gauge) ∗ (
Fill in the blank and explain what this tells us about that operational definition of
rainfall. If that is our operational definition of rainfall, how would it be measured?
11) You should now be able to answer the question: why is height in centimeters (or inches)
a better way to measure rainfall than volume (in cm3 or in3)?
35
AND NOW FOR REAL RAIN GAUGES!!!
Most real rain gauges have another complication that we haven’t looked at yet. This
complication makes it a little harder to understand how rain gauges work, but it makes them
oh-so-much easier to use. We often hear that someplace had 0.03 inches of rain in July.
How do you measure rain every day for a month, add up all of your measurements, and get a
height as small as 0.03 inches? That means on some days the rain must have been less than
0.001 inches – a thousandth of an inch! Could you pour a thousandth of an inch of water into
a soup can? Could you measure a thousandth of an inch with a ruler? How do they do it?
The feature of most rain gauges that makes it possible to measure tiny amounts of rain is that
they are wider at the top than they are at the bottom. A common design for a fancy rain
gauge looks something like the picture below. You should find an example in the classroom.
← Rain does not normally
accumulate above this level
Why make a fancy rain gauge look like this? It actually works better. Let’s think about it.
First of all, you need to know that we only use a gauge like this when the water level is below
the “funnel-shaped” part of the gauge. The exact shape (and volume) of the funnel-shaped
part never enters into our measurement of the rain. (So don’t worry about the funnel-shaped
part! You only need to think about the parts above and below it.)
Second, think back to the “boxes” you made out of little cubes. You made many boxes that
had the same volume but different shapes. Remember? Something similar would happen to
water if you poured it from a soup can shaped like the top of this rain gauge into one shaped
like the bottom.
With that in mind, you should be able to figure out how this rain gauge works and why.
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The “real” rain gauge…
1)
Rainwater accumulates in the bottom of the rain gauge. The volume of the water in
the gauge is proportional to the height of the column of water times the area, but
what area? Lightly draw a column of water in the bottom of the rain gauge and
indicate the height and the area. What area of the rain gauge (times height)
determines the volume of water?
2)
Now imagine that we set the rain gauge in the rain for a few minutes. With our rain
machines we determined that the amount of water that falls into the rain gauge during
that time is directly proportional to the area of the rain gauge, but what area? What
area of the rain gauge determines how much water falls into the rain gauge?
3)
Based on your answer to question 2, what size (and shape) of a soup can would you
need to collect the same volume of rain as this rain gauge? What would have to be
the same about the two rain gauges (the soup can and the fancy rain gauge)?
4)
Imagine that a centimeter of rain falls and we use this fancy rain gauge to measure it.
The volume of rain that falls into the rain gauge is one centimeter times what area?
5)
When that rain falls from the top of the gauge to the bottom of the rain gauge, does
the volume of the water change? Does the (“cross-sectional”) area change or is it the
same as the area from question 4? Does the height change or is it still one
centimeter? What changes and what stays the same?
6)
Volume?
Changes
Stays the same
Area?
Changes
Stays the same
Height?
Changes
Stays the same
Look at your answer to question 5. If all of those quantities are related, and one stays
the same while the other two change, they must change in a very special way. For
each quantity, indicate whether it increases, decreases, or stays the same:
Volume:
Increases
Decreases
Stays the same
Area:
Increases
Decreases
Stays the same
Height:
Increases
Decreases
Stays the same
Explain you reasoning:
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7)
You should be able to tell by looking at the rain gauge that the cross-sectional area
decreases. Let’s imagine that the area at the bottom is only a fifth of what it is at the
top. Remind yourself about the relationship between Volume, Area, and Height, and
then fill in the blanks below as accurately as you can.
Volume:
_______________________________________
Area:
One fifth of what it is at the top.
Height:
_______________________________________
Explain your reasoning:
8)
Take an actual rain gauge and fill it to the level marked “one centimeter” of rain. (If
you have a rain gauge that is “calibrated” in units other than centimeters you may use
10 mm or half of an inch.) Measure the height of the column of water with a ruler.
How tall is it. What does this tell you about the areas of the top and bottom of the
rain gauge?
Pour your “one centimeter” of water into a cup. Save this for later. Now get some
more water and fill the rain gauge up to the bottom of the part where it is widest. The
rain gauge doesn’t have marks to indicate rainfall at this level since we would
normally empty the gauge before it fills this far, but for now fill it just to that level.
← Fill the rain gauge
to this level
9)
Now look back at the water that you poured into a cup. Without actually doing
anything, discuss what would happen if you poured that water back into the rain
gauge with the water level starting at the wide part of the gauge. Think about it and
talk it over. Make a prediction. Be as detailed (quantitative) as you can. Write your
prediction below.
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10)
Try it. Pour the water from the cup into the rain gauge. What happened? Do you
still have that ruler lying around? Check your prediction. Write your observations
below.
11)
Now play. Scientists always save time for play. Try filling the rain gauge up to the
top and pouring water into a cup until the level is a centimeter (or an inch) below the
top. What would you expect to see if you emptied the rain gauge and then emptied
the cup into the gauge? Try it! (It looks cool.) Try holding a rain machine over the
rain gauge and a soup can at the same time. What do you expect to see? What do
you see? Record some observations.
12)
Now try to figure out why they make the rain gauge this way. Imagine 2 cm of rain
falls. If it falls into a soup can, how do you measure it? Could you be sure it isn’t
2.1 cm of rain? 2.05 cm? Fill a rain gauge up to the mark labeled 2.0 cm. Can you
tell the difference between 2.0 cm and 2.1 cm? Could you tell the difference from
2.05 cm? What about the design of this rain gauge makes it easier to make precise
measurements? Explain your answer in words as completely as you can.
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CONGRATULATIONS!
YOU NOW HAVE AN HONORARY DEGREE IN RAIN-GAUGE-OLOGY!
Activity: Make your own rain gauge.
You can make a pretty good rain gauge out of a funnel, a soup can, some modeling clay, a
test tube, and a permanent marker. If you think about it you may come up with your own
design that works even better. Can you figure out a way to make an accurate rain gauge
without the funnel? Are there other designs or materials that would make the rain gauge even
better? Talk to your classmates, work together, design a rain gauge.
YOU CAN DESIGN YOUR RAIN GAUGE IN GROUPS, BUT EVERY STUDENT HAS
TO MAKE A RAIN GAUGE! DESIGN A RAIN GAUGE IN THE SPACE BELOW AND
SHOW YOUR DESIGN TO YOUR INSTRUCTOR FOR APPROVAL. YOU MAY BE
ABLE TO GET SOME SUPPLIES FROM YOUR INSTRUCTOR AS WELL.
IMPORTANT DESIGN FEATURE: The marks on the side of the rain gauge (that say 1.0
cm, 1.1 cm, etc.) are called calibrations. How are you going to calibrate your rain gauge?
How will you find (and mark) the level that corresponds to 1.0 cm of rain? 2.0 cm of rain?
Once you have a rain gauge, take your rain gauge home and record the rainfall for the year!!!
We will plot this data on a spreadsheet with the data from other students to get site specific
detail. Later we will be looking for relationships between this data and other measurements
taken from local rivers and streams. You are now officially a scientist!!!
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EXTENSIONS:
1)
There are people out there who do not want you to be a critical thinker. Advertisers
would almost go out of business if we all thought critically and seriously about what
they were telling us. If a drink really is “the choice of a new generation,” then why
does the new generation have to be told to choose it? A local company claims it will
give you a computer for FREE if you pay $30 a month for the next three years. Your
assignment is to think of four slogans or claims from advertising that might sound
good at first but don’t make much sense when you think about them critically. You
can get them from TV, radio, newspapers, or the web (absolutely no degrading or
offensive material will be accepted!). For each of the four slogans or claims…
a)
b)
2)
Write the slogan or claim as completely as possible.
i)
Describe where you found it.
ii)
Describe what you think the advertiser is trying to make us believe.
iii)
Explain why the claim doesn’t make sense when you think critically.
Do the same for the next slogan, and so on.
Critical thinking, conventional wisdom, and popular myths: There are many things
that we are convinced we know because we have heard them since childhood. Most
people in this country are certain that Christopher Columbus sailed to the Western
Hemisphere in 1492. Still, how do you know it wasn’t 1482? After all, you weren’t
there, were you? You probably know because you were told it was 1492.
As for the year that Christopher Columbus sailed, we have no reason to doubt that
the year was 1492. As critical thinkers we can ask ourselves if that year seems
reasonable and since nobody is trying to convince us that it happened in any other
year, we have no reason not to believe it. The key step in critical thinking is checking
to see whether something is reasonable. When critically considering some statement
or idea, you should ask yourself a few things:
i)
Is it even possible? If somebody tells you they have counted the sides
of a circle and that there are seventeen, you know there is a problem.
ii)
Is it ridiculously unlikely or illogical? If somebody told you not to read
the Sunday edition of the newspaper since the extra pages are really
thin and flat space aliens, you would probably disagree.
iii) Is it reasonable to believe that somebody knows whether this is true?
Suppose a tree fell in the middle of a forest with no people or creatures
or machines around to hear it. If some guy told you that the tree made a
sound like “Gesundheit,” you should wonder how he knows.
iv) If this is true, what else must be true, and could that be true? If you
were told that all do
What follows is a list of ideas that are believed by many people to be true. For each
idea, take a moment to think critically about it, ask yourself the questions above, and
decide whether you believe it. For each one, describe the thought process you went
41
through and explain the reasoning that led to your answer. (There are really no right
and wrong answers here. The reasoning is what is important.)
a)
It is impossible to make any massive object (like a rock or a grain of sand)
move with a speed greater than the speed of light (300 million meters per
second).
3)
b)
Red cars are pulled over for speeding tickets more often than cars of any
other color, just because they are red.
c)
Most of our drinking water comes from “groundwater” that is located in huge
underground lakes in hollow areas underground.
d)
A penny dropped from the top of a skyscraper would make a hole six feet
deep in the sidewalk when it hit the ground.
e)
If all of the people in India jumped into the air and landed at the same time
the Earth would be knocked off course from its current orbit.
f)
Many medical doctors actually know the cure for cancer and have known it
for years, but they aren’t telling anyone about it since they make so much
money off of treatment and research.
g)
It is impossible for any plant or animal to lift a column of water more than
about 35 feet.
h)
It is impossible to divide an angle into three identical smaller angles using
only a compass and a straight-edge (that is without using a protractor or a
ruler or any markings on the straight-edge).
Find somebody wearing “flat” shoes such as sneakers or other shoes with thin soles
and low heels (no heel is even better). Call this person your “standard” (you can be
your own “standard” if you want). Now ask five students to measure the height of
your standard to the nearest millimeter. They should all use the same meter-stick for
measurement, but they must do their measurements independently in a way that they
choose themselves (they should not watch the other students as they measure the
height tester). The height-standard should sit down in a different place before each
measurement so that each student comes up with his or her own place and method for
measuring height. Students should measure the height of the height-standard with
shoes on and then again with shoes off. Record both measurements where students
cannot see. Collect five measurements for the height of your standard in shoes and
five measurements for the height of your standard in stocking-feet (or barefoot).
a)
Make a table that shows the five height measurements with shoes in one
column and the five heights without shoes in the other column.
b)
Look at the measurements of the person with shoes. Are they all the same?
How much do they vary? Are some of them very different from the others?
Use words to describe the variation within this column.
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4)
c)
Look at the measurements of the person without shoes. Are they all the
same? How much do they vary? Are some of them very different from the
others? Use words to describe the variation within this column.
d)
If the columns weren’t labeled, would you be able to tell which column was
the one “with shoes” and which column was the one without? How?
e)
Now imagine a sixth student hands you a measurement that she had made of
your standard, but she does not tell you whether or not the standard was
wearing shoes when it was made. Would you be able to tell whether the
standard was wearing shoes? How? How confident would you be in your
answer? Aside from asking the sixth student about the shoes, what might you
do to become more confident in your answer?
What does it mean to say a container of water is “full?” You might have a mental
picture (an operational definition?) of a full barrel of water, a glass of water, or a
spoonful of water, but what about a penny of water? Here is an experiment that
might surprise you. You need a clean and dry penny, a medicine dropper (or eye
dropper), some water, patience, and steady hands.
a)
Lay the penny on a flat surface. Gently place a small drop of water right in
the middle of the penny. Does the water flow off of the penny? Describe
what you see and try to come up with a hypothesis about why the drop stays
on the penny. (Your hypothesis may be completely wrong. That’s okay.)
b)
Carefully add another drop of water right on top of the first one. What do you
see? You know you put two drops on the penny, but how many droplets of
water do you see on the penny? Again describe what you see.
c)
Imagine adding another drop. Then another. Predict how many drops of
water you can place on the penny before water runs off of the sides. Write
down your prediction.
d)
Now try it. See how many drops you can get on the penny without having
water spill over the side. If water spills over, write down your results, dry the
penny, and try again. You and your classmates can try this many times to see
who can get the most water to stay on the penny. Count the drops. How did
your prediction hold up? Can you do more than you predicted? How much?
e)
Take your best total for the number of drops you got to stay on the penny and
figure out how much water that is in milliliters. One way of doing this would
be to take your dropper to a small measuring device to measure the volume of
100 drops and then use proportional reasoning to get the volume of a single
drop. You could then use that to figure out how much you got on the penny.
f)
Think about the volume of water that you balanced on a penny that you
probably didn’t think would hold any water at all. Would this have any effect
on the amount of water you can carry in a glass? Discuss how important you
think this is as a source of error in measurements of volume.
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5)
Visit or call a carpet store and find out how much 100 square yards of inexpensive
carpet would cost. Then find out how much 200 square yards would cost. Does it
cost twice as much? Suppose one of your classmates argues that since we are talking
about area, it should cost four times as much. Your classmate says, “Surely the cost
quadruples if we double something, but I’m not sure what it is.” Write a couple of
paragraphs and draw a couple of graphs that might explain to your classmate what is
wrong with this thinking and show what doubles as cost quadruples.
6)
Most of us have experienced the phenomenon that when we climb mountains here on
Earth, the temperature decreases the higher we go. We will discuss why this happens
later in this course, but for now we will simply discuss the size of this effect.
The data in the table below was taken on a winter day at and around a forest service
observation tower. The temperature at the top of the tower at 6:00 AM was 0° F, and
all other measurements were taken at different altitudes below that tower.
Distance below tower
100 ft.
Temperature at 6:00 AM
300 ft.
1.7° F
400 ft.
1.8° F
800 ft.
4.6° F
1300 ft.
6.9° F
1900 ft.
10.6° F
0.6° F
a)
Investigate the hypothesis that there is a direct proportion at work here.
Using whatever techniques you choose from the exercises you have done,
make a complete and informed evaluation of the statement that “these
distances and temperatures can be well described by a direct proportion.”
Be as complete as you can, present all of your evidence, and carefully explain
your reasoning.
b)
Whatever your conclusion from part a, come up with your best estimate (or is
it a prediction?) of the temperatures at points with the following altitudes, and
in each case, briefly explain your reasoning.
i)
200 ft. below the tower.
ii)
700 ft. below the tower.
iii) 1000 ft. below the tower.
iv) 2500 ft. below the tower.
c)
Which of your answers in part b are examples of interpolation? Which are
examples of extrapolation?
d)
Estimate the temperature 1000 ft. above the tower and briefly explain your
reasoning.
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7)
When we say that the weather is “humid” we mean that there is a lot of evaporated
water in the air. Search the library, the WEB, or the supplementary textbooks for this
class, and try to find out how much water is stored in the air when there is “100%
humidity” (which is also known as the “saturation point”). The amount of water in
the air is often reported as “grams of water per kilogram of air.” The amount of
water that the air can hold changes with temperature. Try make or find graphs of
water content versus temperature and decide for yourself whether the amount of
water in the air at the saturation point is proportional to the temperature.
8)
On your way to class you pass a couple of calculus students in the hall arguing about
the results they got on a homework problem. The problem was to find the area and
circumference of an ellipse, which is a circle that has been stretched out or squished.
Even though the students used advanced mathematics to calculate the exact answer,
your knowledge of units, length, and area can help them settle their argument.
a)
Since an ellipse is not a perfect circle, the radius in one direction is longer
than the radius in another direction. For this problem, they use the symbol r
for the radius in the short direction. The radius in the long direction is three
times as long, so it is 3r. A picture of the ellipse is shown below:
r
3r
b)
Circumference:
i)
The first student says that he figured out that the circumference is 9π
πr 2.
ii)
The second student says she figured out that it is 6π
πr .
iii) Based on what you know about units, length, and area, what can you
tell them to help them decide which answer might be correct. Explain
your answer.
c)
Area:
i)
Now the first student says he is sure that the area of the ellipse is 3π
πr 2.
ii)
The second student says she calculated the area and came up with 9π
πr .
iii) Again, use your knowledge of units, length, and area to help them
decide which answer might be correct. Explain your answer.
d)
Now you can really impress them. We know that the circumference of a
circle is 2π
πr and that the area of a circle is πr 2. If you look at the picture
above and use proportional reasoning, you can determine exactly what the
area and circumference of that ellipse actually are. What are they? Explain
your answer (and explain why proportional reasoning should apply in each
case).
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9)
10)
11)
Volume: How would you find the volume of a rock?
a)
Take some clay that sinks and cut a 2 cm cube (quick: what’s the volume?).
Completely fill a container with water and gently place the cube in the water,
collecting the water that overflows. How much water flowed over the top?
b)
Take the same cube of clay and roll it into a ball. Refill the container and
place the ball of clay into the water. How much water flowed over the top?
Would it matter if you rolled the clay into a “pancake” or a “rope?”
c)
Find a small rock with an odd shape. Estimate the volume of the rock. Then
repeat the procedure with the container of water, but this time find the
volume of the rock. Does it agree with your estimate? Why or why not?
Work through the area of a circle problem. Look at how it worked its way from what
we know (area of a rectangle) to what we didn’t know (area of a circle).
a)
Can you come up with a formula for the circumference a regular pentagon (or
hexagon) as a function of “radius”? Is it clear what is meant by radius? Take
radius to be the distance from the center of the shape to the middle of an
edge. Now find the relationship between the circumference of the pentagon
and the area. You may do this experimentally, but you should find the same
relationship to be true for more than one pentagon.
b)
Now that you have the relationship between circumference and radius, write
it in the form: circumference =2!(number)!(radius) , and find that number.
You can call it “Judy” if you like. Is “Judy” the same as “Fred” was for a
circle?
c)
Figure out how to construct a rectangle that has the same area as your
pentagon. You can’t make your triangles too small here, so cut a bunch of
triangles the same size and then cut one of them in half. Now you should be
able to assemble the pieces into a rectangle. What are the length and width of
the rectangle? What is the area?
d)
What is the area of a pentagon as a function of radius.
e)
Construct teaching materials similar to the ones used in this class that you
could use to teach a group of middle school or high school students this
method for finding the area of a pentagon. You just went from the area of a
circle to the area of a pentagon, but can you think of a way to get your group
of students to generalize their pentagon to a formula for the area of a circle?
Precipitation and Snowfall. What are the differences between rain and snowfall?
You now know what an inch of rain it, but what does an inch of precipitation mean?
Is an inch of snow the same as an inch of precipitation? Devise an experiment YOU
COULD DO to figure out how to convert “inches of snow” into “inches of
precipitation.”
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12)
Graphing
13)
Spreadsheets
14)
15)
47