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```									                 Indirect Measurement Labs-An Overview

Many times in science, as well as in daily life, objects are not measured using direct methods. The
first lab included in this unit, INDIRECT MEASUREMENT, is designed to confront students
with a series of seemingly impossible tasks (measuring the width of a single piece of paper;
measuring the volume of a single drop of water) in order to force them to THINK of other methods
to measure. They are asked not only to find the measurements of objects but to describe the method
they used to measure each item. These tasks serve as an introduction to the need to devise other
ways to measure besides using a ruler, a beaker, a scale or other measuring devices. Teachers may
choose to do the lab as a demonstration and have the class together derive methods to measure, but
our suggestion would be to take the time to allow the students to work in small groups to come up
with their own ideas. If time is limited, a teacher may decide to assign particular objects to
particular groups or classes.

The additional two labs included in this unit may both be completed or a teacher may decide to do
only one of the two labs. Both can be seen as an application of the idea of indirect measurement and
they are independent of each other.

The second lab in this packet, METRIC MEASUREMENT AND SCALE, deals with metric
measurement from the meter all the way down to the nanometer and has a particular emphasis on
measuring microscopic objects. This lab is particularly applicable for a class that will be using
microscopes in their class. The lab begins with an introduction of the metric units of meter,
millimeter, micrometer and nanometer and the relationships among those units. Students will be
required to convert from one unit to the others. The lab then instructs the students about
microscopic magnification and how to determine the field of view for a particular magnification.
The lab asks the students to indirectly measure the size of various microscopic objects such as red
blood cells, diatoms, pollen grains, etc. The teacher could choose to measure whatever objects for
which he or she has slides available. As a further exploration in understanding relative sizes on the
metric scale, the teacher may choose to ask the students to investigate the sizes of various objects
and to estimate the relative size of other objects as indicated in part D.

the idea of scale using maps and rulers. The student is also asked to calculate one of the missing
pieces of information in the formula Distance = Rate x Time if two of the pieces are known. The
questions in this lab are location specific and could be easily be adapted to fit your particular area.
As noted in the text of the lab, specific tasks could be assigned to specific groups if time is a factor.
In addition to applying to the above science class audiences, this lab could be used stand-alone and
is applicable to math classes at all levels from Algebra I on up.

Measurement: Indirect Measurement                                                          p. 1
Indirect Measurement
Purpose:
To use indirect methods to determine the mass, volume, or length of selected objects.

Introduction:
Sometimes it is not easy or possible to measure a quantity directly so other measurements
are made and the desired quantity determined mathematically.

Equipment/Materials:
Pasteur pipette                            cm ruler
10 ml graduated cylinder                   100 ml beaker

Procedures:
In order to complete the tasks assigned from the list below, follow these 4 steps:
a. Describe the method used.
b. State the variables and controls.
c. Construct a data table.
d. Show calculations.

1.       Determine the average mass of 1 drop of water.

2.       Determine the average volume of 1 drop of water.

3.       Determine the thickness of 1 piece of paper.

4.       Determine the diameter of a piece of copper wire.

5.       Determine the diameter of lead shot (Hint: select uniform pieces and line them up in the fold
of       a sheet of paper).

6.       Determine the number of kernels of popcorn (or grains of rice) in a sealed bag.

Measurement: Indirect Measurement                                                          p. 2
Teacher Notes

Lab Purpose:
The purpose behind this experiment can be as simple or as complex as the teacher wishes. Students
can simply measure the items after watching a teacher demonstrated set of procedures. To use a
more inquiry-based approach, allow students to develop their own procedures. Do not inform them
of the equipment available; instead, allow them to decide which procedural method will be best for

Lab Time:
Depends on how fast the students can think on their toes!

Considerations:
The directions for the students are deliberately vague to provide an opportunity for the students to
develop their own methods. It might be interesting to compare data from the different methods.
Some suggested methods are provided in case students require additional guidance. If time is
limited it might be wise to assign each student group only a few of the exercises.

1. Mass of 1 drop of water
 Place a small container on a balance
 Zero the balance
 Add from a Pasteur pipette a definite number of drops of water (at least 10)
 Determine the total mass of the (10) drops
 Divide to find the mass of 1 drop of water

Weight of group removed
   Place a light weight shallow container large enough to hold a Pasteur pipette horizontally on
the balance
   Fill a Pasteur pipette with water
   Place a pipette in the container on the balance
   Reweigh
   Weigh
   Remove a definite number (at least 10) of drops of water from the pipette
   Reweigh the pipette

2. Volume of 1 drop of water
 Place exactly 5.0 ml of water in a 10 ml graduated cylinder
 Fill a Pasteur pipette with water
 Add water drop wise from the pipette to the cylinder counting the drops until exactly 1.0 ml
of water has been added to the cylinder
 Remember to calculate ml/drop by dividing 1 ml by the number of drops added

Measurement: Indirect Measurement                                                      p. 3
3. Thickness of paper
 A ream of paper is 500 sheets.
 If the paper has not been disturbed, measure the thickness of the ream in cm and divide by
500.
 A smaller number of sheets of paper (200) can be used but will have to be counted.
 It is best if the paper has not been separated since that will create air spaces.

4. Diameter of copper wire
 Look up the density of copper
 Weigh the piece of copper
 Calculate the volume of copper (d = m/v)
 Determine the radius (V = r2h)
 Determine the diameter (d = 2r)
 Note: the volume of copper wire can also be determined by water displacement

 Any shot may be used in which the pieces are uniform
 Crease a piece of paper
 Fold or tape the ends of the crease (to keep the shot from rolling off the paper)
 Place a known number (50-100) of pieces of shot in a line in the crease of the paper
 Make sure the pieces of shot are touching but in a single line
 Mark both ends of the line of shot
 Return the shot to its container
 Measure the distance between the marks made in the step above. This distance represents
the sum of the diameters of the number of pieces of shot used.
 Divide the distance by the number of pieces of shot to get the diameter of 1 piece

6. Number of Kernels of popcorn (grains of rice)
 DO NOT open the bag. The mass of the bag of popcorn should be provided by the
manufacturer.
 Weigh a known number (50) kernels of popcorn
 Calculate kernels/g by dividing the number of kernels by the number of grams
 Calculate the number of kernels in the sealed bag
 (kernels in bag = mass of bag in g / (kernels/g)

Measurement: Indirect Measurement                                                   p. 4
Purpose:
To use map scale representation to navigate from/to a destination. Learn how to set up and
solve proportions as well as use the (distance = rate X time) formula and its derivations to calculate
any unknown given the other 2 parameter values.

Introduction:
We have all been on that road trip. Before going, you will need to plan the most efficient
route in order to accomplish our goals of successfully arriving at our planned destination and
get a rough estimate as to how long it will take to get there.

Equipment/Materials:
State map with scale (road atlas) Gazetteer map with scale (county size – Road map)
Inch/Centimeter ruler             Region map with scale (Multi-state size – Road atlas)

Procedures:
In order to complete the tasks assigned from the list below, follow these 4 steps for all three
trips:
a. Determine the Distance of the trip (in miles) using the map, ruler and the
scale factor for that map.
   Find the Start and End points on the map.
   Measure and record the map distance between the start and end
points. You may need to add a bit of distance to account for the
situation where there is not a direct (straight line) route.
   Determine the mile distance. This will involve setting up and solving
a simple proportion using the scale of miles given for that map.
b. Given the above distance and either the average speed or arrival time, you
will need to calculate the remaining varable.
   If given an average speed, you will be able to complete a rough time
estimate to complete the trip. Often times, you will need to know this
in order to determine the time to start your trip to arrive at a certain
designated scheduled time.
   If given the time to complete your trip, you will be able to compute
the average rate of speed.
c. State the variables and controls.
d. Show calculations.

Measurement: Indirect Measurement                                                        p. 5

1. From the State Map, determine the best motor route from South Bend to Ft. Wayne Indiana and
the distance. Determine the time given an average speed of 50 MPH.

2. From the above trip, determine the average speed given a total time of 3 hours. This will
simulate a stop along the way.

3. From the regional map, determine the best motor route from South Bend Indiana to Washington
D.C. and find the distance. Determine the time given an average speed of 60MPH.

4. You decide to stop and spend the night and split the trip into 2 days. What will your average
speed be if you arrive in Washington 36 hours after you depart from South Bend?

5. For the final exercise, you are to plan a canoe trip down the Pigeon River from Howe IN to Scott
IN. Determine the distance of this trip.

6. If your paddle speed is 3 MPH average, at what time will you need to start in order to complete
the trip before sunset at 6:00 PM?       Include an extra hour of slack time for breaks and to
account for possible time needed to get around downed trees in the river.

Teacher Notes

Lab Purpose:
The purpose behind this exercise is to make students aware of the application of maps as scale
models to provide a real world application of scaling and proportional math to solve real questions
involved in trip planning. Also use the relationships of distance, speed and time to derive each
parameter as a variable given the other 2. Students can simply measure the items after watching a
teacher demonstrated example. To use a more inquiry-based approach, allow students to develop
their own procedures. You can use the provided trips or custom tailor trips for your location.

Class Structure:
Have students perform in groups of 3. Assign roles of Measurer, Calculator and Table Recorder for
each trip. Note that measures may differ a bit from group to group but should be fairly close to each
other.

Activity Time:
35 – 45 mins.

Considerations:
As the distance will be done and is the basis for solving the other questions, the teacher will
demonstrate a specific example (Example: South Bend to Chicago). The formula Distance = Rate X
Time formula should also be provided with related examples for calculating speed and time.
Students will need to do guided discovery in order to interpret the specific questions, set up, and

Measurement: Indirect Measurement                                                      p. 6
solve for each situation. . If time is limited it might be wise to assign each student group only a few
of the exercises.

1. State map - Distance from South Bend to Ft Wayne:
Map measure aprox 4.5 inches taking into account no direct route. From scale, determine
that 1 inch represents 16.5 miles. 16.5 X 4.5 = 74.25 or approximately 75 miles.
2. Average Speed for a 3 hour trip from South Bend to Ft Wayne:
Formula D T  R where D=distance, T=Time and R=Rate of Speed 75 3  25 MPH
3. Regional map - Distance from South Bend to Washington DC:
Measure approximately 4 inches. Scale is 150 miles per inch. 4.5 X 150 = 600 Miles.
Given an average speed of 60 MPH, time would be D R  T 600 60  10hours
4. Average speed given 36 hours to take trip from South Bend to Washington DC.
R = D / T. 600 Mi 36hr  16.67 MPH Average speed with an overnight stop.
5. County Scale Map - Canoe trip Distance Howe to Scott:
The distance from Howe to Scott is 3 inches. You may want to add another inch to account for
the curves in the river. Using the scale of 2.5 miles per inch 4 X 2.5 = 10 Miles.

6. Determine Starting time for Canoe trip.
First the time from the trip is D R  T or 10 3  333Hours. Add an hour for 4.33 hours.
.
Working backward from the 6:00 takeout goal, counting back 4 hours 20 mins would indicate a
starting time of no later than 1:40 PM.

Measurement: Indirect Measurement                                                       p. 7
Metric Measurement and Scale
Purpose:
Measuring and calculating diameter of field of view for the compound light microscope in
order to estimate the size of cells.
Introduction:
In biology and chemistry, many of the objects studied are smaller than the eye can detect.
Therefore, one must use an extension of our senses to even “see” these objects.

Equipment/Materials:
 Transparent centimeter plastic ruler
 Compound light microscope with 40X, 100X, and 400X magnifications
 Prepared permanent microscope slides of various cells. Possible cells are red blood
cells, diatoms, pollen grains, parameciums, amebas.
 Blank slides and coverslips

Procedures: A – Converting from one metric unit to another

Use the following conversions to complete the chart below. Your teacher may give you

meters                  millimeters          micrometers            nanometers
1m                      1,000 mm             1,000,000 um           1,000,000,000 nm
1.0 x 100 m             1.0 x 103 mm         1.0 x 106 um           1.0 x 109 nm
.000000001 m            .000001 mm           .001 um                1 nm
1.0 x 10-9 m            1.0 x10-6 mm         1.0 x 10-3 um          1 x 100 nm
1mm
3.2 x 105 um
75m
48nm
5 um

Measurement: Indirect Measurement                                                    p. 8
Procedure B – Measuring and calculation the diameter of the field of view under the
compound light microscope

When viewing images under the compound light microscope, the diameter of the field of view
changes with the magnification of the microscope. Like a zoom camera, under lowest
magnification, your field of view is relatively large, but all your objects appear very small. As you
increase the magnification, the field of view becomes smaller, but the objects you are viewing
appear larger as you zoom into a smaller area. The diameter of the field of view is inversely
proportional to magnification of the microscope. This means that as the magnification increases, the
diameter of the field of view decreases. Algebraically, this relationship can be written into an
equation. If m1 = one magnification and d1 = the field of view at that magnification, and m2 =
another magnification and d2 = the field of view for that magnification, then

d1 . m1 = d2 . m2

This means that if you know the diameter of the field of view for one magnification, then you can
calculate the diameter of the field of view for any other magnification.

1. Using a transparent centimeter ruler, measure the diameter of the field view in millimeters
while viewing your compound microscope under 40X magnification.

Diameter =     _________mm

2. Since 1 mm = 1000 microns(um), how many microns is your diameter you measured?

Diameter = _______um

3. Using the inverse proportion equation for magnification and diameter, calculate the field of
view diameter in microns for 100X and 400X magnifications. Record your result in the
table below as well as your results from part two:

MICROSCOPE MAGNIFICATION                      DIAMETER OF FIELD OF VIEW
40X                                                um
100X                                                um
400X                                               um

How does the above chart of magnification and diameter of field of view illustrate an inverse
proportional relationship?

_____________________________________________________________________

_____________________________________________________________________

Measurement: Indirect Measurement                                                      p. 9
Procedure C. Indirectly measuring the size of objects under the compound microscope

1. Now that you know the diameter of the field of view under each of your microscope’s
magnifications, you can use that information to estimate the size of objects you examine with your
microscope. For example, if you estimate that 10 cells will fit side by side across a field of view
with a diameter of 2000 um, then each cell will be 1/10th of the diameter or about 200um each. In
the space below, write the equation that can be used to estimate cell size in microns:

Cell size =
_____________________________________________________________________

2. Obtain prepared slides of various organisms and practice estimating their lengths or diameters
using the equation from part 1. Use the magnification that makes it easiest for you to count. In the
chart below, record the name of the cells or object you observe and its estimated length or diameter
in microns.

Name of cell or object                        Estimated size in microns

Measurement: Indirect Measurement                                                     p. 10
Procedure D – Understanding metric scale.

Just how small is a micrometer or a nanometer? One way to understand is to compare the units with
something you can understand or measure yourself. If a micron is one millionth of a meter, then
there are 1 million microns in a meter. Another way of saying this is a meter is a million times
larger than a micron. Likewise, if a nanometer is a billionth of a meter, then there are a billion
nanometers in a meter. Therefore a nanometer is a billion times smaller than a meter.

So how small is a micrometer or a nanometer? To get a good idea, it helps to make some
comparisons with things we understand. For example, lets begin by looking at time. If one second
represents something on the nanoscale, then something on a regular scale would be 1 billion
seconds, which is equivalent to about 31.7 years! To compare a micro with a nano, you need to
understand the microscale is 1000 times larger than the nanoscale, so that is like comparing one
second with 1000 seconds, which corresponds to 16.7 minutes. Looking in the chart below, you
can see this time comparison.

For this activity, you are to make comparisons of things you understand so that it is easy to visualize
the difference between a measurement on a regular scale with one on a microscale and nanoscale.
To do this, you will need to do some measurements and research. For example, in the chart below,
comparing the earth to a marble demonstrates how regular scale compares to nanoscale. To find a
good micro comparison, you need to come up with an object that is 1000 times larger than a marble.
How would you go about figuring that out? Likewise, in an earlier lab, you indirectly measured the
volume of a drop of water. If that drop represents a nano, how what volume would a micro be?
What volume would represent regular scale? Can you relate that volume to an object you can
visualize like a 50 gallon fish tank or an Olympic size swimming pool? When come up with that
object, write it in the chart. Do your best to complete the chart below.

Regular scale                   Micro scale                  Nano scale

31.7 years                      16.7 minutes                 1 second

earth                                                        marble

volume of drop of water

thickness of piece of paper

mass of a drop of water

red blood cell

E. coli bacteria

HIV

Thickness of
One human hair

Measurement: Indirect Measurement                                                      p. 11
Teacher Notes

Lab Purpose:
In biology and chemistry, many of the objects studied are smaller than the eye can detect.
Therefore, one must use an extension of our senses to even “see” these objects. Light microscopes,
electron microscopes, and atomic force microscopes are examples of devices that allow our senses
to be extended to allow us to “see” very small objects. These objects are measured in units of
micrometers or, if very small, in nanometers. A micrometer(micron) is one millionth of a meter or
one thousandth of a millimeter. Nanometers are even small units. A nanometer is a billionth of a
meter or a thousandth of a micron. In this lab you will learn to make metric conversions and to
measure indirectly the size of very small objects.

Lab Time:
2-4 days, depends on if procedure D is done in class.

Objectives:
 Converting meters to millimeters to microns to nanometers and visa versa
 Measure the diameter of the field of view in a light microscope under various magnifications
 Indirectly measure the size of cells under the light microscope
 Measure and calculate a scale of comparison for the units of meters, micrometers, and
nanometers

Possible Extension: Procedure D –Understanding Metric Scale
 Have students investigate the measurements of these items online or by other methods
 Wide range of expansion for take home project
 This ties in the Indirect Measurement Lab

Teacher Notes

For procedure C, you can choose any kinds of objects for the students to measure. For example,
how thick is human hair? Doing that might help them relate scale to cell size.

Also for procedure C, you can demonstrate how cells are counted and size estimated using an
overhead projector, the top or bottom of a standard petri dish, and an assortment of objects like
coins or paper clips. With the petri dish on the overhead projector, place a number of objects on the
petri dish and ask the students how many of the objects will fit across. You can then move the
objects side by side to see what is the best answer. Also, you can use an overhead pen to draw very
small “cells” on the petri dish and then ask students to estimate how many fit across. When they
complain (and they usually do) that the cells are too small too count, help them to figure out that
they can estimate how many fit ¼ or ½ way across and then adjust.

Measurement: Indirect Measurement                                                     p. 12
Measurement: Indirect Measurement   p. 13

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