2.7: Measuring with Metric Units
Introduction
The Javelin
Karina’s seventh grade class is reading a book about Greek history. Karina is fascinated by the Greek Gods and with all of the mythology surrounding them. The class has decided to focus on the sporting events of the Greek Gods since often the stories of mythology are studied in fifth or sixth grade. Their teacher, Ms. Harris thinks this will be a good focus for the class.
“The first javelin was thrown in 708 BC by Hercules, the son of Zeus,” Ms. Harris said at the beginning of class.
“The javelin was originally 2.3 to 2.4 meters long and weighed about 400 grams. Then later, the weight and length of the javelin changed,” Ms. Harris stopped lecturing and began scanning through her notes.
“Hmmm.. I can’t seem to find the place where I wrote down the new dimensions of the javelin,” She said. “Alright, that will be your homework. Also, figure out the difference in length and weight between the javelin of the past and the present.”
Karina hurried out of the class and during study hall began scouring the library for information. She found a great book on track and field and began reading all of the information.
She discovered that the new length of the javelin is 2.6 meters and is 800 grams in weight. Now she needs to figure out the difference.
Measuring metrics and performing operations with metric measurements is what this lesson is all about. It is perfect timing for Karina too. At the end of the lesson, you will be able to help her with her homework.
What You Will Learn
In this lesson, you will learn the following skills:
- Identify equivalence among metric units of measure.
- Choose appropriate tools for given metric measurement situations.
- Choose appropriate metric units for given measurement situations.
- Solve real-world problems involving operations with metric units of measure.
Teaching Time
I. Identify Equivalence Among Metric Units of Measure
Throughout this chapter, you have been working with powers of 10. Let’s review some of the places where you have seen powers of ten. Our place-value system—on both the left and right sides of the decimal point—is based on powers of 10. This fact helps us manipulate numbers so they are easier to use in operations. When adding, we add tens to the left place-value, and when subtracting we borrow tens and regroup them to the right place-value.
In dividing decimals, we multiply the divisor and dividend by 10 until the divisor is a whole number. In scientific notation, we use powers of 10 to convert numbers to a simpler form.
This brings us to measurement and the metric system. The metric system of measurement is also based on powers of 10.
The metric system includes units of length (meters), weight (grams), and volume (liter).
Look at the metric chart below to get an idea of the base-ten relationship among metric units. There are many decimal places, but you will get an idea of which units of measure are larger and which are smaller. This will help you as you learn about equivalence, or about determining which values are equal.
Notice that the metric system has units of length, weight and volume. Our customary system does too, but familiarizing yourself with the metric system is helpful especially when traveling or when working in the sciences.
Metric Units of Length
\begin{align*}&\text{millimeter}\ (mm) && .1 \ cm && .001 \ m && .000001 \ km\\
&\text{centimeter}\ (cm) && 10 \ mm && .01 \ m && .00001 \ km\\
&\text{meter}\ (m) && 1000 \ mm && 100 \ cm && .001 \ km\\
&\text{kilometer}\ (km) && 1,000,000 \ mm && 100,000 \ cm && 1000 \ m\end{align*}
Metric Units of Mass
\begin{align*}&\text{milligram}\ (mg) && .1 \ cg && .001 \ g && .000001 \ kg\\
&\text{centigram}\ (cg) && 10 \ mg && .01 \ g && .00001 \ kg\\
&\text{gram}\ (g) && 1000 \ mg && 100 \ cg && .001 \ kg\\
&\text{kilogram}\ (kg) &&1,000,000 \ mg && 100,000 \ cg && 1000 \ g\end{align*}
Metric Units of Volume
\begin{align*}&\text{milliliter}\ (ml) && .1 \ cl && .001 \ l && .000001 \ kl\\
&\text{centiliter}\ (cl) && 10 \ ml && .01 \ l && .00001 \ kl\\
&\text{liter}\ (l) && 1000 \ ml && 100 \ cl && .001 \ kl\\
&\text{kiloliter}\ (kl) && 1,000,000 \ ml && 100,000 \ cl && 1000 \ l\end{align*}
You aren’t. Remembering all of that isn’t realistic. However, you can learn the prefixes of each measurement unit and that can help you in the long run.
milli - means one-thousandth;
centi - means one-hundredth,
kilo - means one thousand.
Notice that the first two represent a decimal, you can tell because the “th” is used at the end of the definition. The prefix kilo means one thousand and this is not in decimal form.
Now let’s look at some equivalent measures.
Write down these notes before continuing with the lesson.
How can we convert different units of measurement?
You can move back and forth among the metric units by multiplying or dividing by powers of 10.
To get from kilometers to meters, multiply by 1,000.
To get from meters to centimeters, multiply by 100.
To get from meters to millimeters, multiply by 1,000.
Working backwards, to get from kilometers to meters, divide by 1,000.
To get from meters to centimeters, divide by 100.
To get from meters to millimeters, divide by 1,000.
To get from kilometers to millimeters, divide by \begin{align*}1,000,000 \ (1,000 \times 1,000)\end{align*}
Example
Fill in the blanks with the equivalent measurement.
100 centiliter = ___ liter
10 centimeters = ___ meters
1 kilogram = ___ centigrams
1 milligram = ___ centigram
To figure these out, look back at the conversion chart and at the operations needed to convert one unit to another. Whether you are multiplying or dividing, you will need the correct number to multiply or divide by.
There are 100 centiliters in 1 liter.
There are 100 centimeters in a meter. We divide by 100. 10 divided by 100 = .1
10 centimeters = .1 meters
There are 1,000 kilograms in a gram. We multiply \begin{align*}1 \times 1,000 = 1,000 \ grams\end{align*}; there are 100 centigrams in a gram; \begin{align*}1,000 \times 100 = 100,000\end{align*}
1 kilogram = 100,000 centigrams
There are 10 milligrams in a centigram \begin{align*}1 \div 10 = .1\end{align*}
1 milligram = .1 centigram
Notice that when we go from a smaller unit to a larger unit, we divide. When we go from a larger unit to a smaller unit we multiply.
Example
Convert 23 kilograms into grams.
We start by noticing that we are going from a larger unit to a smaller unit. Therefore, we are going to multiply. There are 100 grams in 1 kilogram. There are 23 kilograms in this example.
\begin{align*}23 \times 1000 & = 23,000\\ 23 \ kilograms &= 23,000 \ grams.\end{align*}
Example
Convert 25,000 meters into kilometers.
First, notice that we are going from a larger unit to a smaller unit, so we need to divide. There are 1000 meters in 1 kilometer.
\begin{align*}25,000 \div 1000 &= 25\\ 25,000 \ meters &= 25 \ kilometers\end{align*}
2R. Lesson Exercises
Convert the following units to their equivalent.
- 10 centigrams = _____ grams
- 1 meter = _____ millimeters
- 15 kilometers = _____ meters
Take a few minutes to check your answers with a peer.
II. Choose Appropriate Tools for Given Metric Measurement Situations
Measurement is the system of comparing an object to a standard. As we have seen, the metric system includes units of length (meters), weight or mass (grams), and volume (liter). When we make a measurement of length, weight, or volume, we are comparing the object against a standard (1 meter, 1 gram, 1 volume).
Tools for metric measurements provide these standards.
The metric ruler is the tool for measuring length, and width. It looks like any other kind of ruler but includes units of millimeters, centimeters, and meters.
The balance or scale is the appropriate tool for measuring weight or mass in grams. It looks something like a seesaw. With the object being measured on one side, a combination of standard weights of milligrams, centigrams, and kilograms comprise the other half. When the weight of the object being measured equals the combination of standard weights the scale balances.
A graduated cylinder measures volume. It looks like a drinking glass that has marks for volume on the side—milliliters, centiliters, and liters.
Now we can look at choosing the best tool to measure different objects. You have to think of the object and unit of measure that will be used to measure the object.
Example
Choose the appropriate tool for making the following measurements.
a) The weight of a golf ball
b) The height of a person
c) The volume of water in a chemical solution
d) The width of a table
Now, let’s think about each item and which tool would be the best choice for measuring.
a) The weight of a golf ball would be measured by a scale.
b) The height of a person would be length, so we would use a metric ruler or metric tape measure for this measurement.
c) The volume of water would be measured by a graduated cylinder.
d) The width of a table would be length, so that would be a metric ruler or metric tape.
Thinking about what is being measured whether it is length, weight or volume can help you in selecting the correct/best tool.
III. Choose Appropriate Metric Units for Given Measurement Situations
To find an accurate measurement, it is important to choose the right tool, but it is equally important to choose the right unit of measurement. You wouldn’t want to measure the volume of a swimming pool in milliliters—the number would be too high! The goal in choosing a unit of measure is finding the standard that most closely matches the measure of the object. This procedure is somewhat subjective. It helps to have some real-world measurement benchmarks to act as your own personal standard. Take a look at the following examples.
Length:
- A grain of sand is about 1 millimeter long
- A paperclip is about 1 centimeter wide
- A table could be about 1 meter tall
- A road could be about 1 kilometer long
Weight:
- A speck of dust could weigh about 1 milligram
- A paperclip weighs about 1 gram
- A kitten could weigh about 1 kilogram
Volume:
- An eyedropper holds about 1 milliliter
- A juice box holds about 25 centiliters
- A soda bottle holds about 1 liter
- A kid’s pool could hold about 1 kiloliter
Now let’s practice. Choose the correct unit of measurement for the following objects:
2S. Lesson Exercises
- A bag of flour
- The length of a table
- The amount of water in a pitcher
Take a few minutes to check your work with a friend.
IV. Solve Real-World Problems Involving Operations with Metric Units of Measure
We can combine problem solving and metric measurement with real-world problems. When we have an item that has been measured in metrics, and we need to add, subtract, multiply or divide, then the operations are combined with measurement. Let’s look at an example.
Example
Imogene is buying material for a costume. She buys 3.15 meters of purple material at $4.12 a meter, 2.25 meters of yellow material at $8.63 a meter; and .5 meters of gold ribbon at $1.30 a meter. How much does she spend on her costume? (Be sure to round your prices to the nearest hundredth.)
This problem asks us to find the total Imogene spent on her costume. We know the measurements of the three materials and the cost of each type of material, so we need to multiply all three measurements by their cost and add them together.
\begin{align*}&3.15(\$4.12) + 2.25(\$8.63) + 0.5(\$1.30)\\ &\$12.98 + \$19.42 + \$0.65\end{align*}
Imogene spent $33.05 on her costume.
Next, we can look at an example that uses formulas, measurement and operations together.
Example
Camilla’s brownie pan has a length of 32 centimeters, a width of 16 centimeters, and a volume of .004096 \begin{align*}meters^3\end{align*}. What is the height of Camilla’s cake pan? [Remember: the formula for volume is \begin{align*}V = lwh\end{align*}.]
The problem asks us to find the height of Camilla’s brownie pan. We know the length and width, as well as the area, so we can substitute those values into our equation and solve for the width.
But first—what do you notice about the measurements? The length and width are in centimeters, but the volume is in meters. Let’s begin by converting the volume to centimeters. One cubic meter represents the volume of a \begin{align*}1m \times 1m \times 1m\end{align*} cube. Since there are 100 centimeters in every meter, one cubic meter is equivalent to \begin{align*}100cm \times 100cm \times 100cm\end{align*} or \begin{align*}1,000,000 \ cm^3\end{align*}. To find the volume of Camilla’s brownie pan we multiply \begin{align*}.004096 \ m^3\end{align*} by 1,000,000. \begin{align*}.004096 \times 1,000,000 = 4,096\end{align*}. The volume in centimeters is \begin{align*}4,096 \ cm^3\end{align*}. Now we can solve.
\begin{align*}V & = lwh\\ 4,096 & = 32(16)h\\ 4,096 & = 512h\\ \frac{4096}{512} & = \frac{512}{512} \rightarrow \text{Inverse operations divide both sides by} \ 512\\ 8 & = h \end{align*}
Remember to put the units of measurement in your solution!
The height is equal to 8 cm.
Now we can go back to our original problem!
Real Life Example Completed
The Javelin
Reread the original problem and underline any important information. Then help Karina with her problem.
Karina’s seventh grade class is reading a book about Greek history. Karina is fascinated by the Greek Gods and with all of the mythology surrounding them. The class has decided to focus on the sporting events of the Greek Gods since often the stories of mythology are studied in fifth or sixth grade. Their teacher, Ms. Harris thinks this will be a good focus for the class.
“The first javelin was thrown in 708 BC by Hercules, the son of Zeus,” Ms. Harris said at the beginning of class.
“The javelin was originally 2.3 to 2.4 meters long and weighed about 400 grams. Then later, the weight and length of the javelin changed,” Ms. Harris stopped lecturing and began scanning through her notes.
“Hmmm.. I can’t seem to find the place where I wrote down the new dimensions of the javelin,” She said. “Alright, that will be your homework. Also, figure out the difference in length and weight between the javelin of the past and the present.”
Karina hurried out of the class and during study hall began scouring the library for information. She found a great book on track and field and began reading all of the information.
She discovered that the new length of the javelin is 2.6 meters and is 800 grams in weight. Now she needs to figure out the difference.
To figure out the difference, Karina needs to subtract the old length from the new length. The old length was between 2.3 and 2.4. Karina decides to find the average of the range of numbers. To find the average, you add the values and divide by the number of terms in the series
\begin{align*}2.3 + 2.4 = 4.7 \div 2 = 2.35\end{align*}
Next, she subtracts 2.35 from the present javelin length.
\begin{align*}& \quad 2.6\\ & \underline{- 2.35}\\ & \quad \ .25\end{align*}
There is a difference of \begin{align*}\frac{1}{4}\end{align*} of a meter or .25 of a meter.
Next, we can figure the weight difference.
400 grams was the old weight and 800 grams is the new weight. The new weight of the javelin is twice the weight of the old javelin.
Karina finishes her homework and makes a note of the fact that while the length of the javelin didn’t double that the weight of it did!
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Equivalence
- means equal.
- Metric System
- a system of measuring length, weight and volume
- Milli
- means \begin{align*}\frac{1}{1000}\end{align*}
- Centi
- means \begin{align*}\frac{1}{100}\end{align*}
- Kilo
- means 1000
- Measurement
- comparing the quality of an object against a standard based on what you are measuring.
- Metric Ruler
- the tool for measuring length
- Balance
- the tool for measuring weight or mass.
- Graduated Cylinder
- the tool for measuring volume
Technology Integration
Conversion with metric lengths
James Sousa, Metric Unit Conversion
James Sousa, Example of Metric Unit Conversion
Other Videos:
http://www.mathplayground.com/howto_Metric.html – This is a video on how to explain the Metric system.
Time to Practice
Directions: Fill in the blanks with the equivalent measurement.
1. 1,000 centimeters = ___ meters
2. 10 kiloliters = ___ centiliters
3. 1,000 milligrams = ___ centigrams
4. 100 milliliters = ___ centiliters
Directions: Fill in the blanks with the equivalent measurement.
5. 200 milligrams = ___ kilograms
6. 20 centimeters = ___ meters
7. 2 liters = ___ kiloliters
8. 2,000 centigrams = ___ kilograms
Directions: Fill in the blanks with the equivalent measurements for 180.76 centimeters.
9. ___ meters
10. ___ millimeters
11. ___ kilometers
Directions: Fill in the blanks with the equivalent measurements for 0.4909 kiloliters.
12. ___ liters
13. ___ centiliters
14. ___ milliliters
Directions: Choose the appropriate tool for the following measurements.
15. the volume of a water balloon
16. the length of a basketball court
17. the weight of an apple
18. the volume of a milk carton
Directions: Choose the appropriate unit for the following measurements:
19. the distance between two towns
20. the weight of a peanut
21. the length of a hand
22. the volume of a raindrop
Directions: Solve each problem.
23. The town square has a length of 39.2 meters and a width of 17.5 meters. What is the perimeter of the square? [Remember: \begin{align*}P = 2l + 2w\end{align*}]
24. A pharmacist mixing cough syrup mixes 1,550 milliliters of syrup, 725 milliliters of cherry flavor, and 1.25 liters of water. Once mixed, what is the total volume of the cough syrup in liters?
25. A stable houses horses with the following weights. Arnaud: 331.6 kg; Josie: 331,061 g; Max: 331,612 g; Taboo: 331.61 kg. Order the horses from greatest to least weight.
26. At the farmer’s market, Josh bought 1.5 kilograms of oranges, 150 grams of grapes, and 15,000 centigrams of apples. How many grams of fruit did he buy in all?