TIMS Laboratory Investigations
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(ISBN 0-7872-4084-2, primary): This is a basic investigation that deals with several fundamental concepts in an informal way. Major ideas include the notions of volume, counting and grouping in tens, addition and subtraction, bar graphing, variables, and controlled variables. The children fill a small cup with two different types of beans. The type of bean is the manipulated variable; the number is the responding variable. The size of the cup is controlled. In counting the beans, the children determine which bean is “bigger,” how many more of one type of bean they have than the other, how many cups a given number of one type of bean would fill, etc. This seemingly simple investigation, which only requires filling a cup with some beans, explores a variety of mathematical and scientific concepts that remain important throughout the children’s school careers and beyond.
Picture This—Volume I
(ISBN 0-7872-4085-0, primary & intermediate): The children construct
objects out of Hex-A-Link cubes based on front, side, and top
projections of the objects that are given in the lab. The objects
have different shapes and different volumes. The children determine
the volume of each shape by counting unit cubes. One goal of the
investigation is to develop spatial skill, another is to gain
an appreciation that three dimensions are involved when dealing
with volume, and a third is to begin to get the idea across that
one cannot really tell an object’s volume from knowing merely
one or two of its dimensions.
What’s My Shape?—Volume
(ISBN 0-7872-4086-9, primary): The children pick up 32 Hex-A-Link
cubes and make four shapes of equal volume. They then draw the
shapes using a TIMS cube model plan and determine the length,
width, and height of each object. The investigation has four important
goals: first, to get across to the children that just because
you know the volume of an object does not mean you know its shape;
second, to give the children a feel for the size of volume in
unit cubes; third, to establish the idea that volume deals with
3 dimensions; and fourth, to show that a given volume can have
many different values for each dimension. These ideas are explored
in the comprehension questions. The children conclude the lab
by building models of everyday objects (erasers, etc.) out of
Hex-A-Links and use their model to estimate the volume of the
original object.
Marshmallows vs. Containers
(ISBN 0-7872-4087-7, primary): This investigation makes a direct
assault on the conceptual problem that children and adults have
of associating volume with just the height of a container. Modeled
after an early Piagetian investigation, three containers are chosen
that give results that should confound the children’s expectations.
The children fill the three containers with marshmallows and count
the number in each. Surprise, surprise, the tallest does not hold
the most marshmallows, while the shortest does. Confusion reigns.
This one investigation will not turn them into believers of conservation
of volume, but it is a beginning, and along with the use of inductive
logic, it helps the children to make the step from the particular
situation with marshmallows to the general idea of volume.
Measuring Volume
(ISBN 0-7872-4088-5, primary & intermediate): In Measuring Volume,
the children meet up with the graduated cylinder for the first
time. The purpose of the exercise is to teach them how to use
the graduated cylinder to measure volume, and to become more familiar
with the metric unit of volume: the cubic centimeter. The children
will have a chance to pick things they want to measure, begin
to get a feel for how much volume objects have, and learn how
important conservation of the volume of water is in determining
an unknown volume. Both addition and subtraction are used extensively
as is two-step logic in solving simple but challenging problems.
The graduated cylinder is a piece of equipment the children will
use extensively in subsequent labs, so a carefully done investigation
here will have long-lasting benefits.
Volume vs. Shape
(ISBN 0-7872-4089-3, primary): When children and university students
mash down a piece of clay, they often tell us that it now has
less volume. Indeed, children and adults often confuse area with
volume as well as thickness with volume. Volume is three-dimensional,
and one must look at both area and thickness to get a quantitative
feel for an object’s volume. Believe it or not, you cannot change
the volume of an object merely by changing its shape. The purpose
of our investigation Volume vs. Shape is to help convince the
children that this is the case. In this Piagetian exercise, the
children are given a single piece of clay and are asked to find
its volume when formed into four different shapes. The children
draw each shape and use the graduated cylinder to find each volume,
which, amazingly enough, is the same each time. There are lots
of questions designed to see whether the children really believe
that volume is independent of shape.
Picture This—Volume II
(ISBN 0-7872-4090-7, primary & intermediate): We now move to cubic
centimeters. The students are given cube model plans and construct
the objects out of interlinking cubic centimeters. They then use
the construction to determine the volume of each object in cc.
We continue to explore the idea that volume deals with three dimensions
and whether one can compare the volume of two objects if one only
knows their length, only their width, or only their height, or
sometimes all three dimensions. (No, in all cases.) The children
also learn that knowing an object’s volume does not tell us what
its shape is.
Fill ’er Up!
(ISBN 0-7872-4091-5, intermediate): Fill ’er Up! is a very fundamental
exercise in volume for children. The graduated cylinder is used
as a measuring device to determine the volume of three different
containers, one small (100 cc), one medium (200 cc), and one large
(450 cc). The children have to use adding and subtracting to obtain
the volumes, and in answering the comprehension questions the
children have to multiply, divide, take a ratio, use two-step
logic, and deal with percent. Because we want the children to
get a feel for just how many cubic centimeters of space are in
a given container, we challenge them to predict the volume of
a mystery jar and then determine how close their prediction was.
We also want them to continue to come to terms with the basic
idea that the volume is independent of shape, and that the tallest
container need not always have the most volume.
Volume vs. Number
(ISBN 0-7872-4092-3, intermediate): Volume vs. Number is one of
the important early investigations. The children deal not only
with the concept of volume and how to use a graduated cylinder,
but also with the graphical analysis of data whose best-fit curve
is a straight line that does not go through (0,0). The investigation
explores the relationship between N, the number of marbles in
the cylinder, and V, the volume reading of the water plus marbles.
The manipulated variable is N, and the responding variable is
V. The size of the marble is fixed. The students then repeat the
investigation using a 3/4-inch sphere. The data from both investigations
is plotted on one graph, which makes direct comparisons much easier.
The questions explore the relationships between the slopes of
the lines, their intercepts, and the size of the spheres and the
initial amount of water. Proportions and two-step logic problems
are an integral part of the investigation.
Candle Burning I
(ISBN 0-7872-4093-1, intermediate & middle): We have designed
Candle Burning I primarily as a volume investigation. However,
it can be used as a springboard to discuss gases in general, the
composition of air in particular, and the chemical process of
burning called combustion. The students invert three glass jars
(small, medium, large) over a burning candle and measure the extinction
time of the flame vs. the volume of the jar. What is remarkable
about the data is that the extinction time is proportional to
the volume of the container. Hence we have lots of proportional
reasoning questions for the children. As we discuss in the TLD,
the children can also learn about flames, their sizes, colors,
and the combustion products. And if one is so inclined, one can
add a nice historical note and go more deeply into the meaning
of combustion and a discussion of molecules. You might say Candle
Burning I is a hot experiment.
Evaporation I
(ISBN 0-7872-4094-X, intermediate & middle): This investigation
tries to teach the children something about evaporation, namely,
which variables determine just how quickly evaporation takes place.
The children fill two jars of different diameter with the same
volume of water. They mark the water level every other day for
two weeks, and then using a graduated cylinder, determine the
volume of water in the jar vs. time. The data from both jars is
plotted on the same graph. Besides dealing with a curve that does
not go through (0,0), the children now have to confront a negative
slope. Still, they can interpolate and extrapolate because the
curve is a straight line. The children also use proportions, two-step
logic, and basic algebra to solve a variety of evaporation problems.
They are also asked to design an investigation to see what happens
when one of the controlled variables is changed. There are several
in Evaporation I, so a whole set of paralleled investigations
can be carried out.
Volume vs. Material
(ISBN 0-7872-4095-8, intermediate): The children use a graduated
cylinder to measure the volume of one-inch-diameter spheres made
of steel, glass, plastic, and wood. The manipulated variable is
the type of material, and the responding variable is the volume.
Shape and diameter are controlled. Much to the student’s surprise,
all four objects have the same volume. This important idea is
generalized to other shapes, materials, and dimensions in the
comprehension questions.
Picture This—Volume III
(ISBN 0-7872-4096-6, intermediate): This will be the children’s
third look at projective geometry, as well as a further study
of volume using cubic centimeters. This investigation is more
difficult than the other two because it will combine the use of
small cubic centimeters, three independent projections, and more
difficult shapes to visualize. Once again, we shall home in on
the idea that one cannot always tell the volume of an object by
its outer dimension and that volume deals with three dimensions.
Tower Power I
(ISBN 0-7872-4097-4, intermediate & middle): Using a given floor
plan (i.e., cross section), the children construct towers of different
heights (H) out of cm cubes. For each tower they count cc to determine
its volume (V) and plot V vs. H. The resultant straight line is
used to make predictions and to set up proportional reasoning
problems. Through analyzing their data, the children discover
that the ratio V/H when reduced to a unit common denominator is
simply the area (A) of their floor plan. From this they derive
and use the general equation V = A x H for rectangular right solids.
Tower Power II
(ISBN 0-7872-4098-2, intermediate & middle): The children study
the properties of right cylinders with circular cross sections.
As in Tower Power I, the two primary variables are the volume
of the cylinder and its height. The students use a graduated cylinder
to determine V for three dowels and then plot V vs. H. Once again
the curve is a straight line passing through (0,0). After making
predictions using proportional reasoning, the children explore
the relationship between V, H, and the area (A) of the cross section
of the dowels and eventually derive the equation V = A x H. To
find A, the children can either count sq cm or, if they have done
the investigation Counting Out πR2, calculate the area. They generalize the results to stacks of
washers and then to irregular cross sections.
Candle Burning II
(ISBN 0-7872-4099-0, intermediate & middle): We saw in Candle
Burning I that the extinction time of a candle is proportional
to the enclosed volume of air. Hence, something in the air is
being used up. In Candle Burning II, we try to find out what fraction
of the air is composed of that something. The students invert
a container over a burning candle that is standing in a shallow
pool of water. The water rises up. Using volume, measuring techniques,
proportional reasoning, and percentages, the children determine
just how much of the air was used up. The children make the measurements
for 3 different-sized containers so as to confirm their results.
The lab contains a brief historical discussion of the roles Lavoisier
and Priestley played in the discovery of oxygen.
Volume vs. Diameter
(ISBN 0-7872-4100-8, middle): In this investigation, the children
investigate the relationship between the diameter of a sphere
and its volume. The children have to use all their length and
volume measuring skills as they work with spheres of various sizes.
They are told to be as accurate as possible, which means using
several identical spheres and averaging to find volume and diameter.
The children use spheres of two different materials and plot all
their data on the same graph. The resultant curve of volume vs.
diameter is not a straight line but is independent of the material.
Questions on interpolation and extrapolation follow. The children
then straighten out the curve and find Volume, V, is proportional
to D3. More questions follow on proportional reasoning and taking cubes
and cube roots. A challenging investigation but with math techniques
and insights that will be invaluable in future studies.
Evaporation II
(ISBN 0-7872-4101-6, middle): In Evaporation I the children studied
the evaporation of water over time and saw, qualitatively, that
the slope of the curve of V vs. t depended on the surface area
of the water, A. In Evaporation II the children see whether there
is a simple quantitative relationship between A and the rate of
evaporation, i.e., change of volume divided by change of time.
Three jars of different diameter are filled with water, and the
change in volume of water is determined for each after 5 days.
The best-fit curve of area, A, vs. Dv/ Dt is linear through (0,0) but with a negative slope. Questions
dealing with ratios, proportions, linear reasoning, and multistep
logic sharpen the children’s analytical skills. In an open-ended
addition, the children are asked to find out what happens when
either the type of liquid or temperature is changed. Class discussion
of the applications to drip agriculture and water loss in general
will give the children insight into the value of science in society.
Surface Area vs. Volume or Why Is the Fly Dry?
(ISBN 0-7872-4102-4, middle): Have you ever wondered why elephants
have big ears or why Arctic animals are bigger than their southern
cousins? The answer lies in the relationship between surface area,
SA, and volume, V; between the storage of water and energy and
their dissipation. As we shall see in this investigation, the
balance of surface area and volume will determine the strategy
of survival for plants and animals. The children measure the length,
surface area, and volume of three different-sized cubes. They
will study surface area vs. length, surface area and volume, and
finally, they will study the compound variable surface area divided
by volume versus length. All in all, there are lots of opportunities
to work on nonlinear curves and to look at an inverse proportional
relationship. It is this later relationship between length and
surface area divided by volume that determines many of nature’s
survival tactics.
Mission Impossible—Finding t of a Towel
(ISBN 0-7872-4103-2, intermediate & middle): This is an open-ended
investigation where children are challenged to “find the thickness
of a paper towel using a 100-cc graduated cylinder and an eyedropper.”
The children have to understand how to find the volume of a drop
of water, how to find the area of an irregularly shaped surface,
and understand how to use the relationship V = A x H. Questions
are asked about the frequency distribution of class data, the
most likely thickness, the average thickness, and the thickness
vs. type of towel.
I’m All Wet
(ISBN 0-7872-4104-0, intermediate & middle): This experiment is
designed to review most of the volume concepts the children have
learned so far and to learn a little bit about what humans are
made of. The children are asked to calculate their volume by approximating
each body part with a geometric figure. This means treating an
upper arm as a cylinder, a head as a sphere, a torso as a right
rectangular solid, etc. Where they can, they check their predictions
by the method of displacement. In Part I of the investigation
the students use the volume measurements along with mass measurements
to calculate their overall density and the density of their foot-leg
combination. They use these densities to estimate the amount of
air and the amount of fat in their bodies. What we find is that
we are mostly water, but that fat is an important component that
is burned to supply energy. In Part II the children see if they
can find another variable other than mass that might correlate
with volume.
A Drop in the Bucket
(ISBN 0-7872-4105-9, intermediate & middle): For the first time,
we quantitatively explore the world of molecules and atoms. In
order to do so, the children will need a solid grasp of the concepts
of length, area, volume, and mass, and some of the important relationships
between these variables like V = AL, A = πR2, M/V = constant, etc. The children release a single drop of an
oleic acid, methyl alcohol solution onto the center of a small
amount of powder floating on water and watch the oleic acid spread
over the surface. Knowing the volume of oleic acid and the area
over which it spreads, the children can calculate the length,
L, of an oleic acid molecule. By collecting class data, the children
make a frequency distribution of L and determine the best value
and the error. We discuss in the lab a molecular model of oleic
acid and use the model to find the size of the atoms that make
up oleic acid and the value of their mass.
Lung Capacity
(ISBN 0-7872-4106-7, intermediate & middle): Students measure
lung capacity by blowing air into an inverted container of water
and measuring the volume of water displaced. In this open-ended
investigation, the students pick various manipulated variables
(height, weight, foot size, etc.) and see which one correlates
best with lung capacity. The variables are weakly correlated in
this biological investigation so we have to establish a way of
quantitatively comparing how well each manipulated variable is
correlated with volume. Whole class data is collected to learn
about the distribution of lung capacities, and each student determines
how close he or she is to the average, although there is nothing
average about this interesting investigation.
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Copyright © 1997 by Kendall/Hunt Publishing Company
Copyright © 1999 Institute for Mathematics and Science Education.
All rights reserved.
UIC—University of Illinois at Chicago