Learning Math: Measurement
What Does It Mean To Measure? Part B: Which Rock Is the Largest? (65 minutes)
Session 1, Part B
In This Part
- Measuring Surface Area
- Measuring Volume
- Measuring Weight
Measuring Surface Area
You can measure a number of attributes of rocks — for example, surface area, volume, and weight. Which of these attributes should you use to determine the largest rock? Let’s collect measurements of each type, looking at surface area first. Note 3
The area enclosing a three-dimensional or solid object is referred to as the surface area. Imagine that a thin skin covers all the surfaces of your rock. How would you determine the size of this skin?
For this activity, you’ll need your rock, a sheet of tinfoil large enough to wrap around the rock, and pieces of grid paper with units of 1 cm2, 0.5 cm2, and 0.25 cm2. You can print this grid paper (PDF – be sure to print this document full scale) if you wish.
Before you begin measuring, estimate the surface area of your rock. (Later, you can compare your estimate with the approximate surface area you’ve measured.)
How could you use the tinfoil to find the surface area of the rock? Why would you use this technique?
What unit will you use? Is there more than one choice? Explain.
By first estimating the number of units in a measure, we are forced to consider the size of the object in relation to a standard unit of measurement. If our estimates and approximations are far apart, then we have to reevaluate the size of the unit we chose. When we repeatedly estimate and measure, we improve our ability to measure accurately. It’s this “measurement sense” that allows us to establish benchmarks for particular measures (e.g., “I know that 2 L is about the size of a common soda bottle, so I’d say that this container holds around 3 L”).
Now take the tinfoil and wrap it around your rock, covering the surface area as best you can. Then superimpose the tinfoil that represents the surface area of your rock on the 1 cm2 grid paper and trace around it. Count the number of squares that are completely covered by the tinfoil. This is your inner measure. Then count the number of squares that are both completely and partially covered by the tinfoil. This is your outer measure. Find the average of the inner and outer measures. You can use that average as the approximate surface area of your rock.
How exact is your measurement? How might using a different unit give you a closer approximation? What else might you do to get a closer approximation?
Measure your rock again, this time using sheets of grid paper with units of 0.5 cm2 and 0.25 cm2. Again, superimpose the tinfoil representing the surface area of your rock on each of the grids, trace around it on each sheet of grid paper, and calculate the surface area.
Now compare your three approximations. In order to do this, you’ll need to use the same units, so convert the first two approximations to units of 0.25 cm2. (Be careful here — look at the grid papers and notice how many small squares equal a larger square.) Note 4
What do you notice about the approximate surface areas using different grids? What conclusions can you draw?
In this video segment, Laura and David measure the surface area of a rock. They wrap tinfoil around it and then approximate its area using grid paper. Watch this segment after you’ve completed Problems B1-B4.
What are some of the difficulties they came across? Did you experience similar or different problems?
You can find this segment on the session video approximately 8 minutes and 12 seconds after the Annenberg Media logo.
One method for determining the volume of irregular objects like rocks uses a technique called displacement. Archimedes (287-212 B.C.E.) is credited with discovering volume relationships. Here’s how the story goes: Once there was a king who suspected that his crown might not be made of pure gold. He brought his problem to Archimedes, a “wise man” of the day. Archimedes pondered the question but didn’t have an immediate solution. Later, as he was taking his bath, he noticed that the displacement of water in the tub was equal to the immersed part of his body. Archimedes leaped from the tub and ran naked through the streets, shouting, “Eureka!” His observation showed him how to solve the king’s problem: Using the displacement-of-water method, he could easily calculate the volume of the crown. By comparing the weight of the crown to a lump of pure gold of the same volume, he found that the crown weighed less — indeed it was not made of pure gold. As the king suspected, the crown was composed mostly of cheaper metals. Through measurement, Archimedes was able to expose the jeweler’s dishonesty.
Before you begin measuring, estimate the volume of the rock. (Later, you can compare your estimate with the approximate volume you’ve measured.)
Take a graduated beaker marked in milliliters (or a measuring cup similarly marked) that is large enough to hold your rock. Fill the container halfway and record the water level.
Note that to measure the volume by displacement, you will need to fully submerge the rock in the water. Displacement will be equal to the amount of space taken up by the rock.
What units are you using to measure the water? Can you use this unit to measure the volume of a solid?
One milliliter is 1/1,000 of a liter. There is an interesting relationship in the metric system that can be used to help you here — namely, one milliliter is equivalent to 1 cm3. We often think of measuring solid objects using cubic centimeters, but because of this special relationship, we can also use milliliters (or liters).
You have probably heard the medical term “cc’s.” When used this way, 200 cc’s is equivalent to 200 cm3 or 200 mL. The medical term refers to units of length and volume, rather than units of liquid measure.
Carefully place the rock in the water, and again note the height of the water. Determine the difference in water heights.
If you found the volume of your rock using both displacement of water and displacement of rice, will the measurements be the same? Why or why not?
Do you think that 1 mL of rice equals one cubic centimeter of rice?
Some people might suggest that we examine the weight of rocks to determine which is the biggest. To measure the weight of your rock, you will need a two-pan balance and a three-arm balance.
|Two-Pan Balance||Three-Arm Balance|
Estimate the weight of your rock. (Later, you can compare your estimate with the approximate weight you’ve measured.)
What information can you gather by using a two-pan balance? Can you determine the weight of your rock with this balance?
If you have access to a two-pan balance, use it to determine the weight of your rock.
How does a three-arm balance scale work? Can you determine the weight of your rock with this balance?
If you have access to a three-arm balance, use it to determine the weight of your rock.
In science, a distinction is made between mass and weight. What do you know about these two terms?
How precise are your rock’s measurements? What might affect the precision of this measurement?
Now that you’ve experimented with several different types of measures, which would you use to determine the largest rock in a group of rocks? Should you use a combination of measures?
Take It Further
There are very interesting relationships among metric measures involving water. One cubic centimeter of water is equivalent to 1 mL of water. In addition, 1 mL of water (or 1 cm3 of water) weighs 1 g. You may then conclude that the amount of water your rock displaced should be equivalent to the weight of your rock. What is faulty about this line of reasoning?
Is the rock heavier, lighter, or the same weight as water?
Are there any other measurements you could use to determine the largest rock in a group of rocks?
Prior to measuring these attributes, it is important to consider what type of unit should be used. Children frequently have a difficult time choosing appropriate units of measure. For example, they often try to measure area using linear units (centimeters), or volume using two-dimensional units (square centimeters). Reflect on your own knowledge of metric units of measure. Does your knowledge of and familiarity with metric units have anything to do with your ability to choose an appropriate unit?
The following equivalencies may be helpful: 1 cm2 = 4(0.5 cm2), and 1 cm2 = 16(0.25 cm2). In order to visualize these relationships (e.g., that four 0.5 cm squares cover the same amount of space as 1 cm2), draw the 0.5 cm squares on 1 cm2. Reflect on or discuss why the average of the inner and outer measures is the approximate surface area and not the exact surface area.
The relationship between cubic centimeters and milliliters (1 cm3 = 1 mL) and, accordingly, between cubic decimeters and liters (1 dm3 = 1 L) will be explored further in Session 3.
This method works best with a beaker marked in milliliters. Other containers may not have adequate markings for you to determine the volume of water displaced when you submerge the rock. If you don’t have such a beaker, you could fill the container to the top, measure the amount of water that overflows when you submerge the rock, and then pour the overflow into more precisely marked measuring devices.
Since spilled water can be messy, you might try using a solid material instead, such as fine sand or rice. Fill a container to the top with sand, place your rock in the container, collect the overflowing sand, and then measure the amount that overflowed.
In this session, we are using the common term “weight,” even though we are technically finding the mass of the rock. The difference between the terms is discussed in detail in Session 3.
Also, many teachers have not had the opportunity to study different types of scales, chiefly because scales are not standard equipment in classrooms. To do a hands-on version of this activity, you’ll need a two-pan balance and a three-arm balance. Using the two-pan balance, you can compare an object on one pan to a set number of weights on the other pan, adjusting the weights until the pans are level (or balanced). The three-arm balance has weights built into the instrument. You may be able to borrow scales from colleagues in middle or high school science departments. Try to use the best scales your school system has available. Be sure that the scales have been “balanced” prior to using them.
Some people may consider the weight of the rock to be an exact amount, perhaps because it is more difficult to think about using smaller and smaller units to measure weight. But is weight ever exact? Reflect on or discuss this problem as a group.
If you and your colleagues are working in several small groups, try to decide which group has the largest rock. Unless the rocks are very different, though, this might not be a simple task. The term “largest” is not an absolute and has many meanings, depending on the circumstances and the judgement of those involved with the decision-making process. Ultimately, you may choose to use a combination of measures, such as those that are discussed in the homework.
You could wrap tinfoil around the rock, just covering it; then unwrap the tinfoil, lay it flat, and measure its area. One reason to use this method is that it is far easier to measure a flat (two-dimensional) area than it is to measure the surface area of a three-dimensional object, and the tinfoil can be laid flat while still representing the three-dimensional surface area. Tinfoil is also quite flexible and can wrap tightly around most irregular surfaces of the rock.
Yes, there is more than one choice. The units for surface area will be square: square inches, square centimeters, square millimeters, and so on. The size of the unit depends entirely on the size of the object being measured. Since the rock you used is relatively small, it is reasonable to use square centimeters or square half-centimeters as the unit of surface area.
It is not very exact. There are several sources of error, including the error of estimating that the tinfoil has exactly enveloped the rock, and the error of rounding our answers. A closer approximation could be obtained by using a finer grid. It is more difficult to overcome the error of estimating using foil, but other materials could be used instead.
Answers will vary, but you will probably find that the approximate areas are different for each type of grid. Again, measurement is not exact and is subject to error, including error introduced by the method of measurement itself.
But more importantly, you will probably notice that the inner and outer measures are closer together on the finer grid paper. This means that your answer will be closer to the actual area. By using smaller units, we can increase the precision of our measurements and get better approximations.
The units are in milliliters. Yes, this unit can be used to measure the volume of a solid because of the relationship between a milliliter and a cubic centimeter. The number of milliliters of displacement will be equal to the number of cubic centimeters of volume for the rock.
Probably not. Since rice is solid, it will not completely fill the container, and several measurements of the same amount of rice will likely have different results. Additionally, since we are measuring twice with two different methods, there is very likely to be a measurement error between the two.
You can determine a measure (i.e., an approximation) for the weight of your rock relative to other known weights, but not the exact weight of the rock.
The three-arm balance works on the lever principle, in which moving a weight farther from a balance point produces a greater force on that side of the balance. (This is the same principle used in balancing a seesaw.) We can determine an approximation of the rock’s weight using this type of scale, but not the exact weight.
Mass is a measure of the amount of material making up an object (specifically, its molecules). All objects have mass, but not all have weight, which is the effect of a gravitational field on a body that has mass. For example, a U.S. flag placed on the Moon has the same mass as one placed on the Earth, but it weighs less as a result of the Moon’s gravitational pull. Objects can be weightless, but they can never be without mass.
The precision is based on how fine the measuring instrument is. In a two-pan balance, precision is based on the values of the pan weights being used. The smaller the value of the unit, the more precise the measurement. For example, measurements made using milligrams are more precise than those using grams or kilograms.
The type of measure you use depends on what you’re really looking for, since there is no absolute meaning of “largest.” In a group of rocks, one may have the greatest surface area, another may have the greatest volume, and a third may have the greatest weight. The meaning of “largest” depends on circumstances and the judgement of those involved in the decision-making process. So, for example, if you decide that the largest rock is the heaviest rock, you would use a scale, rather than the tinfoil-and-grid-paper or water-displacement methods.
The fault in this line of reasoning is that your rock does not have the same density as water. For most rocks, 1 cm3 of rock weighs more than 1 cm3 of water. This can be seen by noting whether the rock sinks or floats when placed in water. If it sinks, it is denser than water and has more weight than the same volume of water.
You can choose to measure length in any direction, once the rock is placed in a particular orientation. Or you might choose to measure the circumference of the rock. As with surface area, volume, and weight, none of these linear measurements could be used by itself to determine if a rock is “largest” — unless that’s your only criterion!
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.