Part 1: Length, Capacity, and Weight

Tell students, “The two measurement systems that we work with are the Metric System and the Customary System. Do you know a key characteristic of the Metric System?” (powers of ten)

“The distinct characteristic of the Metric System is the base 10 foundation for all measurements.”

Divide students into groups of three to four. Have students take a few minutes to research the following question: “Which countries primarily use the Metric System? Which countries primarily use the Customary System?” Discuss and compare group responses:

• Some countries that primarily use the metric system: China, Germany, Japan, India, Indonesia, France, Spain, Sweden, Russia, South Korea, Canada, Mexico, and Brazil.
• Some countries that primarily use the customary system: United States, Great Britain. The additional countries utilizing the Customary or SI system include Myanmar (formerly Burma) and Liberia. In the UK, the Customary System is not recorded as the official measurement system. However, such units are mandatory on distance signs. There is contradictory information about the official measurement system for Myanmar and Liberia. However, much literature shows the Customary System to be the measurement system of choice for these two countries.
• The metric system is preferred throughout the world, due to the foundation of the base 10 system. Therefore, knowledge and conversions are easily transferable from one country to another. The U.S. has made a declaration to use the metric system for all trade and transportation purposes. However, it has been very slow in the process. The customary system is still shown more focus in the elementary mathematics curriculum.

Note: Almost all countries, excluding the U.S., use the Metric System. Thus it is very important that all Americans can convert from the Customary System to the Metric System. However, that investigation is beyond the scope of this lesson. This lesson is only interested in performing conversions within each system. Therefore, we will convert measurements within the Metric System, and also convert measurements within the Customary System.

“Let’s start with Metric System conversions. We are interested in three distinct measurements: length, capacity, and weight.”

“Here is an important question: What is the difference between capacity and weight?” Give students time to discuss the question. (Capacity measures the volume of a container, amount the container can hold, or amount the container does hold. Weight is a measurement of how much something weighs.)

“Before we begin conversions, let’s examine the measurement prefixes we will encounter. What are some common prefixes for the Metric system? You often hear of milli, centi, and kilo, when referring to the Metric System. These prefixes are used with all measurement types.”

“Now, what do you suppose each prefix represents?” (milli = thousandth, centi = hundredth, kilo = thousand)

Distribute the Metric Units Prefixes handout (M-8-5-2_Metric Units Prefixes and KEY.docx). “Let’s review these interpretations in a brief table.” Review the Metric Units Prefixes table on the handout with students.

“We will now perform conversions for each of the three measurement types, starting with length. With length, we will look at and convert between the following:

• meter
• millimeter
• centimeter
• kilometer

Let’s begin with 1 meter and the equivalent conversions for the other three measurements. In your group, take a few minutes to discuss what the conversions will look like and fill in the blanks in the table, knowing the following:

• milli represents
• centi represents
• kilo represents 1000”

“We can formulate a plan to convert meters to the other measurements. First of all, we notice that a millimeter is shorter than a meter. We also notice that a centimeter is shorter than a meter. However, a kilometer is longer than a meter. How can such knowledge help us in the conversion process?” Draw out the discussion so that students recognize the most efficient operation for the conversion. “In other words, what do we do when we convert from a greater number to a lesser number?” (Dividing by a positive number greater than one results in a smaller quotient; dividing by a positive number less than one results in a larger quotient. Similarly, multiplying by a positive number greater than one results in a larger product; multiplying by a positive number less than one results in a smaller product.) “From a lesser number to a greater number?” (Select the most efficient operation according to the conversion factor.)

Give students time to discuss. Then say, “Intuitively, we might think that converting from a greater number to a lesser number involves division. However, that is certainly not the case. Instead, you multiply by a multiple of 10. Why would we multiply instead of divide?” (When a factor is multiplied by a whole number factor of ten, the product will be greater than the factor.)

Ask, “Any ideas? Of course, since there are more of the smaller units within the larger unit, the smaller unit will thus have a greater value.”

“When converting from a lesser number to a greater number, we do the opposite. We divide by a multiple of 10 because there will be less of the greater number within that lesser number.”

“So, in applying this knowledge to our conversions, how could we show the processes for our solutions? Let’s add the solution process for each conversion to our table.” Draw students’ attention to the Metric Unit Conversions table on the handout.

“Now, suppose we look at such conversions in the other direction. The formal naming convention for millimeters, centimeters, and kilometers includes mm, cm, and km, respectively. We have the following equations:”

1000 mm = ____ m

100 cm = ____ m

0.001 km = ____ m

“In your group, provide the process for solving each problem. We already know the solution is 1 meter. The intent here is to show the process for converting from millimeters, centimeters, and kilometers to 1 meter.”

Ask students to share their processes, any questions, discoveries, difficulties, etc. Review the following information with students: “To go from 1000 mm to ___ m, we notice that we are converting from a smaller unit to a larger unit. Thus, we will divide. To solve, we have:

“To go from 100 cm to ___ m, we again notice that we are converting from a smaller unit to a larger unit. Thus, we will, once again, divide. To solve, we have:

“Finally, to go from .001 km to ___ m, we notice that we are converting from a larger unit to a smaller unit. Thus, we will multiply. To solve, we have:

“Now, instead of simply converting from millimeter, centimeter, and kilometer to a meter and vice versa, we will now convert between the other units.”

“Using the information you have learned, fill in the conversion table—the last able on the handout—and provide an explanation for how you determined each value. Note any patterns or consistencies in the table. Is there a resulting pattern that can easily help you convert from a particular lesser number to a greater number and then convert back from that greater number to the lesser number? Are there any similarities?”

“Write a list of conversions in the empty spaces of the table in the handout, revealing your understanding of the conversions. For example, 1 mm = 0.10 cm, and 1 cm = 10 mm.”

“Do these conversions make sense? Are they reasonable, according to size?” Invite discussion. “Also, notice that if 1 km = 1000 m, then 1 m = 0.001 km, whereby 1000 and 0.001 are multiplicative inverses, such as .”

“Now that we have an understanding of the foundation of conversions of length, our discussion must advance further. We must now examine decimal amounts and other amounts, excluding simple multiples of 10.”

Example:

8.19 km = ____ mm

“Using our existing knowledge and the conversion chart, 1 km = 1,000,000 mm. In other words, to convert from 1 km to mm, we multiply by 1,000,000. Therefore, in order to convert 8.19 km to mm, we simply multiply 1,000,000 by 8.19. Doing so gives 8.19 km = 8,190,000 mm.”

“We can set up the following ratio:”

Example:

56 cm = ____ m

“Using our existing knowledge and the conversion chart, 1 cm = 0.01 m. In other words, to convert from 1 cm to m, we divide by 100. Therefore, in order to convert 56 cm to m, we divide 56 by 100, or multiply 0.01 by 56. Doing so gives 56 cm = 0.56 m.”

“We can set up the following ratio:”

Example:

129 cm = ___ mm

“Using our existing knowledge and the conversion chart, 1 cm = 10 mm. In other words, to convert from 1 cm to mm, we multiply by 10. Therefore, in order to convert 129 cm to mm, we simply multiply 129 by 10. Doing so gives 129 cm = 1,290 mm.”

“We can set up the following ratio:”

Hand out to students the Lesson 2 Exit Ticket (M-8-5-2_Lesson 2 Exit Ticket and KEY.docx). Have students complete the exit ticket to assess their understanding before moving on with the lesson.

Capacity (Liters) and Weight (Grams)

Tell students, “In the Metric System, capacity and weight are each measured using the same prefixes of milli-, centi-, and kilo-. Therefore, it is nice to know that you can apply the foundation you just learned to these two types of measurements. The only differences will be the ending portions of the words. Let’s explore the measurement units for capacity and weight.” Ask,

• “Does anyone know the base unit for capacity measurement in the Metric System?” (It is the liter.)
• “Thus, we can convert between milliliter, centiliter, and kiloliter, denoted as mL, cL, and kL, respectively.”
• “What is the base unit for weight measurement in the Metric System?” (It is the gram.)
• “Thus, we can convert between milligram, centigram, and kilogram, denoted as mg, cg, and kg, respectively.”

Alternative Teaching Approach: Using Dimensional Analysis

1. Discuss with students how kilo, milli, and centi relate back to their base unit of gram, liter, meter, etc.
   1. 1000 meters in a kilometer
   2. 100 centimeters in a meter
   3. 1000 millimeters in a meter
2. “Conversions can be performed using a method called dimensional analysis, which applies the concepts of cancelation to get to a desired unit.”
3. Review the previous example: 8.19 km to mm
   1. by multiplying the fractions across and dividing by a common factor leads to 8,190,000 mm.
4. Point out to students that fractions must be set up so that none of the units are found both on top or both on bottom, since the idea is for them to be able to cancel away.

Create an explanatory PowerPoint on the topic “Converting Capacity and Weight Measurements in the Metric System.” Use animation and illustrations throughout. Include an array of varying conversions. Include at least two real-world illustrations and conversions, e.g., converting a liter soda bottle to milliliters.

Customary System Conversions

“Next, you are going to discover the need to convert customary units via a) data sets, and b) data that you personally gather.

With length, we will look at and convert between the following:

• inch
• foot
• yard

With capacity, we will look at and convert between the following:

• fluid ounce
• cup
• pint
• quart
• gallon

With weight, we will look at and convert between the following:

• ounce
• pound

Let’s first look at some conversion examples.”

Example:

“With this example, we simply convert one unit below feet. Therefore, we know there are 12 inches in 1 foot, so we can write:

“Now, let’s look at another example, whereby we must convert two units above a measurement, or two units above inches, in this case.”

Example:

“We know there are 12 inches in 1 foot. We also know there are 3 feet in 1 yard. Therefore, there are 36 inches in 1 yard. We perform the following operations:

Or using dimensional analysis,

Therefore, 122 in. ≈ 3.38 yd.”

“These examples related only to length. Let’s look at a capacity example and a weight example.”

“Suppose we need to know how many cups will hold x amount of fluid ounces. Given that Alisha must use 24 fluid ounces in a recipe, how many cups will she use?”

“We must determine how many fluid ounces there are in 1 cup.” Give students time to respond. “There are 8 fluid ounces in 1 cup. You might check a small milk carton. There are 8 fluid ounces in there. If you pour the milk into a measuring cup, you will find the milk fills the cup to exactly the 8 oz. line.”

“How would Alisha solve the problem?” Give students time to respond. “She would set up a conversion equation, similar to the ones we just finished. She could write:

“Therefore, 24 fluid ounces = 3 cups!”

“Now, let’s look at a weight conversion. Suppose we are asked to determine how many pounds and ounces a freight shipment weighs. This problem is a bit harder. Suppose the package to be shipped weighs 117.6 pounds. We want to know the weight in pounds and ounces.”

“We start out knowing that it weighs 117 pounds plus a certain number of ounces. We need to determine how much of a pound, in ounces, .6 pound really is. Therefore, we simply multiply .6 pound by the number of ounces in a pound, which is 16.”

“Therefore, the shipment weighs 117 pounds and approximately 10 ounces!”

“Now, what if we wanted to know the total number of ounces for the weight? What would we do? Give students time to respond. We would use a conversion ratio and set up an equation like this one:”

“So, the package weighs a total of 1,881.6 ounces. we can check this solution another way, as well. We can start by finding the number of ounces equivalent to 117 pounds and adding the 9.6 ounces for .6 pounds!”

“Then,

As you can see, we found the same solution, or number of ounces!”

Before beginning the conversions of data sets, distribute the Conversions worksheet to students to practice common conversions (M-8-5-2_Conversions and KEY.docx). Students are only expected to convert up to 2 units above or below the given unit, so parts of the table will need to be filled in.

Students often have difficulty distinguishing between fluid ounces for capacity and ounces for weight. In addition, consumers often have difficulty knowing which units are being presented. For example, a soup can is marked with ounces for net weight, not volume/capacity. Consumers expect other liquids to be marked with the cost per volume.

Once students have completed the Conversions worksheet, they should practice conversions using actual data and/or data sets. The Web sites and suggestions in Related Resources at the end of the lesson provide a variety of data and/or data sets for converting.

Provide one data set or data resource for each measurement type. Thus, provide a separate resource for length (inch, foot, yard), capacity (fluid ounce, cup, pint, quart, gallon), and weight (ounce, pound). Remind students about the distinction between ounces that measure weight (mass) and fluid ounces that measure volume.

Student Directions:

a)      Perform conversions for each measurement type, using the given data. You are only required to convert up to two units above or below each unit. However, feel free to do other conversions. (Note: For each data set or resource, make sure that each unit is included. In other words, for the data covering length, include conversions for inch, foot, and yard. For the data set covering capacity, include conversions for fluid ounce, cup, pint, quart, and gallon. For the data set covering weight, include conversions for ounce and pound.)

b)      Choose a measurement type to explore: length, capacity, or weight. Gather data for the measurement type and convert according to a real-world need. Explain why those conversions are important. (Note: Students might choose capacity and decide to gather their own beverage volume information.)

Alternative Method: Using Dimensional Analysis

• Prior to going through examples of conversions, print out a chart of measurement equivalencies and go through with students the conversions that will be necessary to complete Lesson 2. Therefore, students have a chart to refer to while practicing conversions (M-8-5-2_Customary Conversions Chart and KEY.doc).
• Go through the same examples shown in Lesson 2, but use dimensional analysis and the conversions on the chart to solve the problems.
• This method can be simpler for students to follow rather than using the conversion tables because they do not have to use fractions. Also, it provides a more consistent method that is similar for every conversion problem.

Part 2: Time and Temperature

Time

“When discussing the topic of time, what units of measurement do you think about?” Give students time to respond, then say, “We often think of the following:

• Seconds
• Minutes
• Hours
• Days
• Weeks
• Months
• Years”