SCIENCE SAMPLER
Sinking and floating:
Bringing math to the
surface
stores. Pill bottles
or test tubes with
caps can also be
used as long as they
are airtight.
• paperclips or other
small items
• balances
Procedure
The concept of density underlies the explanation for a
variety of natural phenomena such as weather patterns
and plate tectonics. However, we often find that although
students can identify the formula d = m/v, they are
not able to apply it to other concepts in science. Full
understanding of density requires understanding the
mathematical explanation behind it. In these activities,
the mathematics is coaxed into student explanations
of obser ved phenomena. This builds facility with
two impor tant mathematical concepts: ratio and
proportion.
Explore one
Students investigate how changing one part of the ratio
(volume or mass) affects whether or not an object sinks
or floats. In each activity, one variable changes while the
other is held constant.
Question: If we add to or subtract mass from an object,
does it affect whether or not the object sinks or floats?
(Volume is constant.)
FIGURE 1
Mass (g)
Sinker
In this first activity,
Sinker
students will use a film
canister to maintain
Floater
a constant volume,
but change what they
Floater
put inside to alter
the mass. Students
Volume (mL)
begin by recording the
volume of their canister. As the volume remains constant,
it will only need to be measured once. However, be aware
that students may not realize this. Water displacement is
the most accurate method of measuring the volume of the
canister. Once students have decided on what they wish
to place in the canister (coins, paperclips, metal washers),
the mass will need to be measured and recorded. Students
then place the canister in the water and record on a
whiteboard or lab book what occurs. Students draw or
describe how the canister floats or sinks: Is a little bit
sticking out of the water or a lot? Does it sink slowly or
quickly? Students should continue changing the mass
of the canister by changing what they place inside until
they have a few examples of both “sinkers” and “floaters.”
FIGURE 2
Mass (g)
Sinker
Sinker
Floater Floater
Materials (per group of three to four
students)
• chemical splash glasses
• large tub or 1,000 mL graduated cylinder of water
• several empty film canisters. These can be obtained
from any place that develops film, such as drug
Volume (mL)
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SCIENCE SAMPLER
Explore two
FIGURE 3
Question: If we change an object’s volume, does it affect
how the object sinks or floats? (Mass is constant.)
Mass (g)
Potato
Water
Apple
Volume (mL)
Students should have a T-chart with the mass of the
canister and a drawing/description of how the canister
behaved in the water. Sample questions to ask student
groups while you circulate include the following: Describe
what you did. How did you make a sinker float and a floater
sink? Is there a specific point where it is neither a sinker
nor a floater? If so, what is that point?
Explain one
Students graph their data and label each data point (v,
m) to see that each data point represents a coordinate on
the graph (see Figure 1). In the explain phase, students
generate a rule based on their data so far: What makes
something sink or float? Explanation of their rule should
utilize words, pictures, and numbers. It should also
reflect the data that were collected. This rule should
be negotiated in small groups of three to four students.
Large whiteboards per group are perfect for this because
students can erase and revise as needed. If whiteboards
are not available, chart paper or butcher paper will work
as well. When ready, student groups can share their rules
with each other for feedback. Sample student rules are
“Adding mass makes the canister sink,” and “Things with
less mass float and things with more mass sink.” At this
point, students’ rules will reflect the misconception that
heavy things sink. This point will be addressed as students
continue through the lesson. Students will be asked to
refine this rule at each stage of the lesson.
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SCIENCE SCOPE
Materials (per group)
• one set of 500 mL water bottles per group: three
water bottles with the tops cut off at different heights
(2 cm/4 cm/7 cm) to create three different volumes.
Place masking tape across the tops to avoid sharp
edges.
• a tub of water or large graduated cylinder
• weights (rocks, marbles, and so on)
• plastic wrap
• rubber bands
Advance prep: You will need three water bottles for each
group of three to four students. Prior to class, cut off the
tops of the bottles with scissors so that there are three
different heights: small, medium and tall. This should
take about 15 minutes.
Procedure
In this second activity, students place the same mass
in dif ferent-sized (volume) containers. Students
select one mass from the weights listed above (rocks,
FIGURE 4
Where could you plot a point to
represent a possible density?
Mass (g)
Volume (mL)
SCIENCE SAMPLER
marbles) and record it. Students place the selected
mass in each of the three empty water bottles (of
var ying sizes) and obser ve if it sinks or floats by
placing the bottle in a tub of water or graduated
cylinder. In order to prevent water from getting
into the bottle, students cover the bottle with plastic
wrap and secure it with a rubber band. Be careful of
any extra plastic wrap floating in the water, as it will
interfere with the behavior of the bottle. The mass of
the bottle will change with the size and the wrapper.
This difference is negligible but you may want to bring
up this issue, depending on your students’ progress
at this point in the lesson. If the graduated cylinders
or beakers are large enough, students use water
displacement to measure the volume of each bottle. A
less precise method would be to fill the bottle to the
top with water and then pour the water into a graduated
cylinder for measurement. Students should record the
mass, volume, and behavior of each container.
FIGURE 5
All possible densities plotted
Mass (g)
Explain two
Students graph their data and label each data point
(v, m) (see Figure 2). Students return to the rule
they generated earlier, adding to or adjusting the rule
based on their obser vations in the second activity.
As with the previous explain phase, the rules should
be discussed in groups of three to four students, and
changes or adjustments should be written on the same
sheet of chart paper or whiteboard. Sample questions
for small groups: If you could change either mass or
volume, how you could make an object a better floater
or a better sinker? What af fected how well these
cylinders sank or floated? Typical answers include,
“Increasing volume or decreasing mass makes the
cylinder float better.”
Students have now investigated both aspects of
density: the measurement of mass and volume. Density
is the ratio of those two measures (d = m/v). Students
will need to see sinking and floating in terms of this
ratio: The higher the density, the more likely the object
is to sink. How you introduce this formula depends
on where your students are as a class. They may have
been introduced to the concept previously, students can
read about density in their text, or you can explain it to
your class directly. However the formula is introduced,
students now have a context in which to discuss this
ratio. Students can go back to the data that you graphed
in activities one and two and calculate the density of each
Volume (mL)
object. Ask students to consider how this ratio could be
used to restate their rule.
Explore three
Up to this point, students have not needed to compare
densities—only different amounts of masses or volumes.
This third activity requires the comparisons of different
densities with the ratio m/v held constant. Students are
given various objects: apples, potatoes, brownies, water,
butter, chocolate bars, and such. You may wish to remind
students that food items should never be consumed in a
lab. Students should change the quantity of each object
(i.e., cut them into smaller and smaller chunks). If you
are concerned with safety issues you can select items that
are easily cut with a plastic butter knife (butter, brownie).
Students should keep track of the mass and volume of the
various amounts. This can be done in their small groups.
Although each group does not need to investigate every
material, they all need to measure different ratios of water,
as these will be important parts of the graph.
Explain three
Student graph their results (see Figure 3). In this graph,
each line represents the density of an object such that
each “chunk” of substance is represented by a data point
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SCIENCE SAMPLER
FIGURE 6
Mass (g)
Graphing the rules for activity three
Sinker
mw
ms
>
Vw
Vs
Water
mw
mw
=
Vw
Vw
Floater
mw
mf
<
Vw
Vf
Volume (mL)
and these points are all proportional to each other. After
students make the graph for different-sized pieces of
certain objects (e.g., apples and potatoes), the following
discussion is appropriate: On this graph (Figure 4), where
could you place a point to indicate a density? Answer: By
definition, any point on this graph (but not on an axis) is a
specific density. If all possible densities were plotted, then
the resulting graph would resemble the solid block shown
in Figure 5. So, what would a diagonal line mean on this
graph? The diagonal line is all of the points with the same
density. Density of the object is represented as the slope
of each line. This means that the ratios would be equal (in
proportion). Mathematically, the line represents all of the
m
m
points (v1, m1), (v2, m2), such that v 1 = v 2 . Students may
1
2
note that any points that represent neither sinking nor
floating on graphs # 1 and # 2 would be on the water line
from activity # 3 because the ratio of mass and volume at
these points is equal to the density of water.
Students return to the rule they generated earlier, adding
to or adjusting the rule based on their observations in the
last activity. As with the previous explain phases, the rules
should be discussed in groups of three to four students
and changes or adjustments written on the same sheet
of chart paper or whiteboard. The rules should include a
mathematical description of what makes something sink or
float. Questions for small groups: What does the graph tell
you about the object/substance you measured? On a given
line, what is changing as you move along it? What stays
the same? What does the comparison of the lines show
us? How is this graph alike and different than the first two
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SCIENCE SCOPE
graphs? Where are the points on this graph? Graphically,
an explanation or rule one could generate from the third
activity is demonstrated in Figure 6.
A floater in water is an object that has a density less than
that of water. A sinker is an object with a density greater
than water. Graphically, all points above the water density
line represent sinkers and all of those below represent
floaters. Lines that represent various amounts of the same
object will be steeper (greater slope) than water’s line if it
is a sinker and less steep (lesser slope) if it is a floater.
Extend
Change the water to a different medium, such as salt water
or alcohol (rubbing alcohol requires chemical splash
glasses, proper ventilation, and flammability precautions).
Predict and verify what will happen to the objects (apples,
potatoes, and so on) from the explore three when placed
in the new liquid. Salt water is denser than water, so some
sinkers will float in salt water. Alcohol is less dense than
water, so some floaters will sink in alcohol. Graphically,
the line separating sinkers and floaters will be more or
less steep, depending on the density of the liquid.
Conclusion
We often refer to density as if it were a single property
when in fact it is the ratio of two properties: mass and
volume. Whether something sinks or floats is based on
the comparison of this ratio of the fluid and the object. A
mathematical explanation allows one to see the relationship
that exists between all of these variables and ratios, proving
very useful in moving students toward a full understanding
of this complex phenomenon. In this activity we investigate
each piece individually. First we held the volume constant
and varied the mass. Second, we held mass constant
and varied the volume. Finally we held the ratio constant
(mass and volume) and varied the mass and volume
proportionately. This mathematical discussion includes the
measurement of each property (mass and volume), the ratio
of these properties (density), and the comparison of this ratio
between two substances (graphically as slopes).
Susan Gomez-Zwiep ([email protected]) is an
assistant professor at California State University, Long
Beach, and a Cadre member of the K–12 Alliance.
David Harris ([email protected]) is a math coach
with Vista and Escondido school districts in California
and is a regional director for the K–12 Alliance.