Learning Math: Measurement
The Metric System Part B: Metric Units (85 minutes)
Session 3, Part B
In This Part
- Liquid Volume
- Reasoning with Balance Scales
As we’ve seen, the base unit for length (or distance) is the meter. Meter comes from the Greek word “metron,” which means “measure.”
Many of us do not have a strong intuitive sense of metric lengths, which may be a result in part of our limited experience with metric measures and estimates. It is, however, important to have referents for measures, as referents make measurement tasks easier to interpret and provide us with benchmarks against which to test the reasonableness of our measures.
Print this centimeter grid (PDF – be sure to print this document full scale) and paste it onto stiff cardboard paper. Cut and tape pieces together to build a meterstick, and explore how you would mark decimeters, centimeters, and millimeters on it.
- Find a friend or colleague, and use a metric tape measure to measure the following body lengths: A, B, C, D, and E (as pictured below). Your goal is to try to find your own personal referents for 1 cm, 1 dm, and 1 m.
- Using the information you gathered, estimate these lengths:
- The height of a door
- The length of your table
- The width of a notebook
- The thickness of a dime
- The length, width, and height of the room
Take It Further
- Approximate a distance of 100 m. First plan how you will determine this length, and then measure this distance outside. Mark off 100 m using chalk, and then use a trundle wheel to check your approximation. (A trundle wheel is a plastic wheel, usually graduated in 5 cm intervals, designed to measure lengths by counting the number of clicks, each of which equals 1m.)
- Use this distance to estimate the time it would take you to walk 1 km. If it’s a nice day, check your estimate by actually walking 1 km. What is your average walking pace? Note 6
In this video segment, Mary and Susan work together to establish some referents for measuring lengths using their own bodies, for example, the width of a hand or an arm length. They use those referents to make measurements and then compare them to standard-unit measurements.Why is it important to establish such referents for measuring? Can you think of any situations in which they might be useful? You can find this segment on the session video approximately 10 minutes after the Annenberg Media logo.
Measures of liquid volume, sometimes referred to as capacity, include the liter (L) and the milliliter (mL). These terms are holdovers from an older version of the metric system and, because they are so well known, are approved for use with the current SI. Volume, whether liquid or solid, is a measure of space. Solid volume is measured using cubic meters (m3) as the base unit. Liquid volume is most often measured using liters. In Session 8, we will explore measures of solid volume in detail, but we will begin to examine the relationships among measures of solid and liquid volume in this session.
By definition, a liter is equivalent to 1,000 cm3 (or one 1 dm3). This leads to the conclusion that 1 mL is equivalent to 1 cm3. Large volumes may be stated in liters but are usually recorded in cubic meters.
- Use metersticks and masking tape to construct a cubic meter. The metersticks will form the edges of the cubic meter.
- Make a list of measures equivalent to 1 m3 (for example, using cubic decimeters, cubic centimeters, cubic milliliters, and liters).
Estimate the capacity of the following in liters or milliliters:
- A teacup
- A thimble
- A car’s gas tank
Sometimes in medical situations, we hear of someone receiving an injection of 3 cc of medicine. What do you think this measure, 3 cc, represents?
Examine a 1 L bottle. In addition to the liquid, there is some air space in the bottle. So what is a liter — the amount of liquid, or the entire volume of the bottle? Find out by pouring the liquid into a graduated 1 L or 500 mL container. Try this with a 2 L or 3 L bottle as well.
How much liquid is actually in a 1 L bottle? In a 2L bottle?
The average woman has a lung capacity of about 4.4 L, and the average man has a lung capacity of about 5.8 L. What is their lung capacity in cubic decimeters? Compare the two units.
Whereas weight measures the gravitational force that is exerted on an object, mass measures how much of something there is; thus, mass is closely related to volume. The weight of an object can change depending on its location (e.g., on the Earth or on the Moon), but the mass of the object (how much of it there is) always stays the same.
Mass and weight are often confused, because our two systems of measurement use different terms. In the metric system, kilograms and grams are measures of mass, but in the U.S. customary system, ounces and pounds are measures of weight. When using the metric system, we should really state that we are measuring mass, saying, for example “I have a mass of 60 kg” rather than “I weigh 60 kg,” but this goes against convention. Throughout this course, we will use both terms (but regardless of the term we use, mass is what we’ll be finding!).
The base unit of mass is the kilogram (kg). In the 1790s, a kilogram was defined as the mass of 1 L (cubic decimeter, or dm3) of water:
Though that definition has changed somewhat with time, here is a definition that is close enough for ordinary purposes: There are 1,000 g in 1 kg, and 1,000 g occupy a volume of 1,000 cm3, or 1 L. Therefore, 1 g of water weighs the same as 1 cm3 of water and occupies 1 mL of space. In other words, for water:
1,000 g = 1 kg = 1,000 cm3 = 1 dm3 = 1 L
1 g = 1 cm3 = 1 mL
Kilograms are used to weigh just about everything but very light objects (which are weighed in grams) and very heavy objects (which are weighed using metric tons). A gram is almost exactly the weight of a dollar bill. A metric ton is equivalent to 1,000 kg (so it can also be thought of as a megagram) and should not be confused with the common American ton in the U.S. customary system. In fact, the metric ton is often referred to by its French and German name, tonne, to distinguish it as a metric measure. Most cars have a mass of between 1 and 2 tonnes; a large diesel freight locomotive has a mass of approximately 165 tonnes.
As with metric lengths, it is useful to establish benchmarks for metric mass measures.
- .Using the centimeter grid paper (PDF), build a cubic decimeter. What is its capacity? What is its weight if filled with water?
- What is the capacity and weight of 1 cm3?
How many cubic centimeters are there in a cubic decimeter?
Some people use balance scales like the one illustrated above, and some people use spring scales like a typical grocery or bathroom scale. How are the two different?
Think about how they work and whether they measure weight or mass.
Look around the room you’re in now, and find one or more objects that you estimate has each of the masses listed below. If a scale is available, use it to measure each object to corroborate your estimates. If a scale is not available, select some food products (with very light wrappers) such as candy bars and cereal. Estimate their mass and compare your estimate to the mass indicated on the product.
- 1 g
- 100 g
- 500 g
- 1 kg
Take an empty plastic liter bottle and weigh it. Then fill the bottle with very cold water and weigh it again. What do you notice about the weight of the liter of water? Explain your findings.
Watch the participants as they measure the mass of a 1 Liter bottle filled with water. They are surprised to discover that it’s a little more than 1 kg. This is not what they expected, so they contemplate possible explanations of their result.
Can you think of any other possible reasons that would explain why the bottle weighs more than a kilogram?
You can find this segment on the session video approximately 19 minutes and 20 seconds after the Annenberg Media logo.
How will filling the liter bottle with something other than water — for example, juice, yogurt, or sand — affect the mass? Explain.
Think about the relationship between mass and volume. Can two substances have the same volume but different mass?
Estimate the mass of a newborn baby, a fifth grader, an adult woman, and an adult man. If possible, use scales to gather your data.
Reasoning With Balance Scales
Knowing something about mass and how scales respond when objects are placed on them allows us to reason logically about weight or mass. For example, fake coins sometimes make their way into circulation, and balance scales can be used to determine which coins are fake. Let’s explore this further.
Take It Further
- Imagine that you have three coins. One is heavier than normal, so you know it’s a fake. But you cannot tell which one is the heavy one just by looking. What is the minimum number of weighings you would need to complete to find the fake coin?
- How would you find the fake coin among four coins? What is the minimum number of weighings needed?
- How would you find it among six coins? What is the minimum number of weighings needed?Rather than weighing two coins at a time, think about how you could divide the coins into groups and then compare the groups to eliminate some coins.
- How would you find it among eight coins? What is the minimum number of weighings needed? Note 14
- How would you find it among 12 coins? How many weighings would it take?
How is the metric system used in the United States today? Watch this video segment to find out how veterinary medicine uses it to provide the best and safest treatment for its patients.
Do you know of any other professional fields that rely on the metric system?
You can find this segment on the session video approximately 21 minutes and 22 seconds after the Annenberg Media logo.
When measuring objects using the metric system, it is important to establish benchmarks for common lengths, such as meter, decimeter, and centimeter. In addition, you should actually make the measurements and compare your estimates to your measurement data. Reconciling the differences between your estimates and measures will help you improve your ability to make reasonable estimates using the metric system.
Be aware that errors in approximating 100 m are going to be compounded tenfold when using your 100 m distance to approximate 1 km (since 100 • 10 = 1,000 m). You may want to check the distance using a car or bicycle odometer. When you know the approximate time it takes you to walk 1 km on flat terrain, then you can use time to estimate distances (e.g., I walked for 20 minutes, so I know I have traveled about 2 km).
Whereas the base unit for volume is the cubic meter, most practical day-to-day situations find us determining the capacity of smaller containers, and thus cubic centimeters or cubic millimeters might also be used. The relationship between cubic centimeters and milliliters (1 cm3 = 1 mL) and between cubic decimeters and liters (1 dm3 = 1 L) is an important one to establish. Models can help people visualize these relationships. If you have metric base ten blocks, then the “small units cube” (1 cm3) is equivalent to 1 mL, and the “thousands cube” is equivalent to 1 dm3; this cube, if hollow, will hold 1 L. Compare a milliliter and a cubic centimeter as well as a liter and a cubic decimeter. If possible, pour 1 L of water into a hollow decimeter cube.
If you are working in a group, it is worth the time and effort to have groups of four people construct a cubic meter, using metersticks as the edges of the large cube. If supplies are limited, make 1 m3 as a model for the whole group to observe. Participants can hold the metersticks in place or tape them together.
Some people may have difficulty listing the equivalent measures for 1 m3. A common error is to think that there are 1 cm3 in one cubic meter. Use the model to show that this amount is much too small. Try listing an equivalent measure for each dimension before finding the volume. For example, since 100 cm = 1 m, the length, width, and height of the cube are all 100 cm, and you can find the volume by multiplying length times width times height.
Since you will be pouring liquids in and out of bottles to find the capacity, this can get a bit messy. But to fully understand the size of metric units, it is important to actually do the measurements. If there is time, test more than one type and brand of liquid. Is there the same amount of liquid in a 1 L bottle of soda as in a 1 L bottle of water? Or test many 1 L bottles of the same brand. Is the capacity consistent from one bottle to the next? Be sure to discuss or reflect on your findings.
If you are working in a group and you did not discuss the difference between mass and weight during Session 1, do so now. Have everyone explain in their own words how mass and weight differ. Ideally you want to have available a variety of scales: pan balances, three-arm balances, spring scales, and a metric bathroom scale.
If you are working in a group, when finding materials that have masses of approximately 1 g, 100 g, 500 g, and 1 kg, work individually. You can use more than one object to reach the target mass: Place the objects in a plastic bag and then label the bag with the combined mass. As a group, discuss each quantity, answering questions like “What does a mass of 1 g feel like?” and “What common items have a mass of 1 g?” Each group member should examine the different samples and then weigh them again, using the different scales.
If you do not have a metric bathroom scale, you can use a conversion factor to change pounds to kilograms. Since 1 kg = 2.2 lb., you can find someone’s weight in kilograms by dividing his or her weight in pounds by 2.2.
This activity focuses on reasoning deductively about mass, but it does not further one’s knowledge of metric measures. The problems, however, do show that there are many ways to measure without using units.
You might be tempted to solve this problem in a similar way to part (c), by first placing four coins on each of the pans. This enables you to conclude that the heavy coin is one of the four (one pan will go down!). You can then make additional comparisons in order to identify the heavy coin. There is, however, another way to solve this problem that takes only two steps. Hint: The first weighing does not involve all eight coins. Can you figure out how to identify the heavy coin?
Answers will vary.
- Answers will vary. Distance A is approximately 1 cm. Distance B is approximately 8 or 9 cm. Distance C is approximately 1 dm. Distances D and E are approximately 1 m.
- Answers will vary. Measure the lengths in terms of your own referent measures and then approximate.
- One way to estimate distances is to know the length of one’s individual pace. First determine your average walking pace, and then use this information to approximate 100 m.
- One way to do this is to walk the 100 m, then multiply the time it took to walk 100 m by 10. Your average walking pace would be measured in meters per second, and could be found by dividing 1,000 by the total time taken (in seconds) to walk the kilometer.
- You will need 12 metersticks to do this, one for each edge of the cube.
- One cubic meter is equivalent to 1,000 dm3, since there are 10 dm in each dimension and three dimensions: 10 • 10 • 10 = 1,000. Similarly, 1 m3 is equivalent to a 1 million cm3 and 1 billion mm3. Since 1 L is equivalent to 1 dm3, there are 1,000 L in 1 m3.
Answers will vary, but here are a few examples:
- A teacup holds about 200 to 250 mL (or 2 to 2.5 dL).
- A thimble will probably be about a tenth of a teacup, or 20 mL.
- A gas tank of a car holds anywhere from 40 to 70 L.
The phrase 3 cc refers to cubic centimeters, which is equivalent to milliliters; “3 cc” of medicine is 3 mL.
If you measure how much liquid is in a 1 L or 2 L bottle, you’d most likely find that there is a small amount of extra liquid in each, probably as a result of a particular bottling procedure.
A cubic decimeter is the same volume measure as a liter. So a woman’s lungs hold about 4.4 dm3 of air, and a man’s lungs about 5.8 dm3 of air.
- The dimensions of the cubic decimeter are 10 • 10 • 10 cm. The capacity is 1 L, and the weight of 1 L of water is 1 kg.
- One cubic centimeter of water weighs 1 g. The capacity of 1 cm3 (or cc) is 1 mL.
A balance scale compares two different masses. It typically only tells us when one side of the scale is larger in mass than the other.
A spring scale uses the force of gravity to determine the weight of an object placed under the spring.
The biggest difference between scale types is that some scales rely on mass calculations (typically balance scales), while others rely on weight calculations (typically spring scales). One interesting thought is whether these scales would report different answers on the Moon; the spring scale would because it depends on gravity, and the balance scale would not.
Answers will vary. Here are some possibilities:
- A large paper clip weighs about 1 g.
- A chocolate bar is generally 100 g.
- A small bag of flour is 500 g.
- Four medium apples together will weigh approximately 1 kg.
Depending on the temperature of the water and the level of precision of the instrument you used, you should notice that the measured mass came close to 1 kg. This confirms what we know already: One liter of water has a mass of 1 kg when the water is at 4 degrees Celsius.
Depending on the object used, the weight may not be 1 kg. For example, sand is heavier than water, so a cubic decimeter of sand (a liter of sand) should weigh more than a cubic decimeter of water. In general, a substance heavier than water but with the same volume weighs more than 1 kg. A substance lighter than water weighs less than the same volume of water.
A newborn baby has a mass of approximately 3 to 4 kg. A fifth grader has a mass of approximately 35 to 40 kg. An average adult woman has a mass of approximately 60 to 70 kg. An average adult male has a mass of approximately 75 to 90 kg.
It takes only one weighing. Put one coin on one side of the balance and a second coin on the other side. If one is heavier than the other, the heavier coin is the fake. If they balance, the third coin is the fake.
It takes up to two weighings. Weigh two coins against each other. If one is heavier, it is the fake one. If they balance, weigh the other two coins against each other. Again, the heavier coin is the fake one.
Alternatively, you could start by putting two coins on each side of the balance. The side with the fake coin will be heavier. Weigh those two coins against each other to determine which one is heavier, and thus fake.
The minimum number of weighings is still two. Put three coins on each side of the balance. The heavier side will contain the fake coin. This reduces the number of possibly heavy coins to the three on that side. Then the method from part (a) can be applied to find the heavy coin.
Alternatively, you could divide the coins into three groups of two. Weigh two of the groups against each other. If one side is heavier, the fake coin is there. If they balance, the fake coin is in the group that was left out. Weigh the two coins in the “bad pair” against each other; the heavier is the fake one.
You can do this with three weighings. Put four coins on each side for the first weighing. The heavier side will contain the fake coin. This reduces the number of possibly heavy coins to the four on that side. Then divide the heavier group into two groups, and weigh them; finally, weigh the remaining two coins from the heavier group.
You can also do this with two weighings: Place three coins on each side of the balance. If they’re the same weight, proceed to weigh the two remaining coins that haven’t been weighed. If not, use the method in part (a) to decide which is the fake coin.
e. It will take three weighings. First, divide the 12 coins in half, weigh them, and determine which half the heavier coin is in. Then with the heavier side, follow the six-coin strategy from part (c).
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.