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# 2.8: Converting Metric Units

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Shot Put

When Marcus was in sixth grade he thought that he was a runner. So, he tried out for the track team and his love of running and determination made him an easy choice for the team. All that first year, Marcus ran. Then, in the hurdles, Marcus fell and hurt his knee. Marcus was devastated. He thought his life of track and field was over.

While his knee healed, Marcus watched his teammates practice. One day after practice, Marcus picked up the shotput and threw it across the field. His coach was watching and Marcus found a new way to participate.

“Sometimes life is like that,” his coach, Mr. Samuels, said. “You think you are going to be doing one thing and something else crosses your path.

This year, Marcus is on the shotput team. His team is trying to figure out which field they should use to practice in. There are several different fields available, but the team is large, so they are looking for a field with a width of 45.65 meters.

Field A—0.004565 kilometers

Field B—45,650 millimeters

Field C—456.5 centimeters

Field D—456,500 millimeters

Which field should his team choose? Marcus is puzzled and you may be too. This lesson will teach you all about converting units of measure and comparing them. By the end of it, you will be able to help Marcus figure out the best option for his shotput team.

What You Will Learn

By the end of this lesson, you will be able to demonstrate the following:

• Convert metric units of measure using powers of ten.
• Compare and order given metric units of measure.
• Estimate equivalence between metric and customary units of measure
• Solve real-world problems involving conversions of metric units of measure.

Teaching Time

I. Convert Metric Units of Measure Using Powers of Ten

As we have seen, the metric system of measurement is based on powers of 10. When you have a measurement in one unit, we convert it to another unit by either multiplying or dividing by some power of 10.

For example, to get from kilometers to meters, multiply by 1,000. Working backwards, to get from kilometers to meters, divide by 1,000. To get from kilometers to millimeters, we divide by 1,000,000 (1,000×1,000)\begin{align*}1,000,000 \ (1,000 \times 1,000)\end{align*}. When converting among measurements, use prefixes as a guide in multiplying or dividing by powers of 10.

You can ask yourself, “Am I moving from a smaller unit to a larger unit, or from a larger unit to a smaller unit?”

If the units you’re converting to are smaller, multiply.

If the units you’re converting to are larger, divide.

Now try to do some conversions without using the conversion chart. If you remember the rules of moving the decimal point, you should be able to do these quite easily. Keep in mind the relationship between the powers of 10 and moving the decimal point to the right or left. Think of it as the operation of dividing making the number smaller, while multiplying makes the number larger.

Dividing by 1,000, for instance, has the same effect as moving the decimal point three places to the left. Similarly, multiplying by 1,000 has the same effect as moving the decimal point three places to the right.

Example

Convert the following measurements to centimeters, 525 meters

There are 100 centimeters in a meter, so to go from meters to centimeters, we multiply by 100 or move the decimal point two places to the right.

Example

Convert the following measurements into kilograms, 95231 milligrams

To move from milligrams to kilograms, we divide by 1,000,000 (1,000 milligrams in a gram ×\begin{align*}\times\end{align*} 1,000 grams in a kilogram), or move the decimal point six places to the left.

Yes. Now it is time to practice what you have learned. Here is a table to help you.

KM to MM to CMCM to MMMM to CMCM to MM to KM×1000×100×10÷10÷100÷1000\begin{align*}&\text{KM to M} && \times 1000\\ &\text{M to CM} && \times 100\\ &\text{CM to MM} && \times 10\\ &\text{MM to CM} && \div 10\\ &\text{CM to M} && \div 100\\ &\text{M to KM} && \div 1000\end{align*}

2T. Lesson Exercises

Practice converting the following units by moving the decimal point.

1. 500 m =_____ cm
2. 120 m = _____ km
3. 50 cm = _____m

Take a few minutes to check your work with a peer.

II. Compare and Order Given Metric Units of Measure

Knowing how to convert between metric units of measure makes comparing and ordering measurements possible.

Do you remember how to compare decimals? You line up the decimal points and compare the place values from left to right. Because units of measure often involve decimals, comparing measurements is similar. But to compare measurements, they have to have the same unit!

Example

Compare 4.56 g to 4.59 g.

First, we have to be sure that these are the same unit of measure. Both are in grams, so we can look at the numbers to determine which is larger. 4.56 is less than 4.59, so we have our answer.

The answer is 4.56 g < 4.59 g.

Example

Compare 743 km to 74,300,000 mm

The first thing to notice is that these two units of measure are not the same. Therefore, we have to convert both of them to the same unit to compare them. We can convert both measurements to kilometers. Multiply kilometers by 1,000,000 to get millimeters.

743×1,000,000=743,000,000\begin{align*}743 \times 1,000,000 = 743,000,000\end{align*}

So we can compare 743,000,000 to 74,300,000.

Our answer is that 743 km > 74,300,000 mm.

Once you understand how to compare units, ordering them becomes quite simple. Just remember that you always have to convert the units so that they are the same!!

2U. Lesson Exercises

Practice comparing metric units of measure.

1. 45 cm _______ 5.5 mm
2. 2 km _______ 400 m
3. 6 l ________ 60,000 ml

Take a few minutes to check your work with a friend. Are your answers accurate? Did you convert to the same unit of measure before comparing?

III. Estimate Equivalence between Metric and Customary Units of Measure

We have been working inside the metric system of measurement.

Did you know there is another system of measurement? The customary system of measurement is widely-used in the United States. You are probably already familiar with some of the customary units. Customary units of length include the inch, foot, yard, and mile; for weight, ounce and pound; for capacity, teaspoon, cup, pint, quart, and gallon.

We encounter both metric units and customary units all the time—sometimes at the same time. One important aspect to understanding measurement is developing a general idea of the equivalence between the two systems.

The chart below shows the estimated metric equivalent to common customary units. Notice how all the customary units are in units of 1. 1 foot, for instance, is about 30.48 centimeters, so 2 feet is about 60.96 centimeters or 30.48×2\begin{align*}30.48 \times 2\end{align*}. If you had a measurement in centimeters and wanted to estimate the number of inches, you would reverse the process, or divide by 30.48.

Customary Units Metric Units
1 inch 25.4 millimeters
1 foot 30.48 centimeters
1 yard 0.91 meters
1 mile 1.61 kilometers
1 teaspoon 4.93 milliliters
1 cup 0.24 liters
1 pint 0.47 liters
1 quart 0.95 liters
1 gallon 3.79 liters
1 ounce 28.35 grams
1 pound 0.45 kilograms

Completing these calculations exactly can be very tricky, so we use estimation to determine equivalence.

Example

How many cups are equal to .48 liters?

To answer this question, we can look at the chart. We can see that one cup is equal to .24 liters. Therefore, we can say that 2 cups is equal to .48 liters.

Example

4 kilograms is equal to how many pounds?

First, let’s look at the number of pounds to kilograms on our chart. 1 pound is equal to a little less than one-half of a kilogram. To find the number of pounds in a one kilogram, we can multiply the pounds by 2. 2 pounds are in 1 kilogram. Therefore, there are 8 pounds in four kilograms.

By estimating, we can find approximate equivalent measures for customary and metric units of measure!!

IV. Solve Real-World Problems Involving Conversions of Metric Units of Measure

Metric units play key roles in measurements for athletes. Some examples of this include: track and field distances like the one in our introduction problem, in the size of an Olympic swimming pool, in the length and width of football fields and basketball courts. But metric units play important roles in many other aspects of everyday life; from the weight of vegetables at the farmer’s market, to the width of living room drapes.

Now that you know how to convert and compare across metric units, you can use this knowledge to examine real-world situations. Before we go back to our introductory problem, let’s try one of these real-world problems.

Example

At the Harvest Festival, Jocelyn is trying to buy the largest pumpkin. Covert the mass of the following pumpkins into the same unit and order the pumpkins from greatest to least.

Pumpkin J—5.67 kg

Pumpkin K—5,510,000 mg

Pumpkin L—567,100 cg

Pumpkin M—5,800 g

This problem asks us to order the pumpkins from greatest to least based on their mass. In order to do so, we need to make sure all the pumpkin weights are being represented by the same measurement unit. We’ll covert them to grams because grams are a good middle-point between the units represented.

• Pumpkin J has a mass of 5.67 kg. To convert to grams, we need to multiply by 1,000 or move the decimal point three places to the right. 5.67 kg \begin{align*}\rightarrow\end{align*} 5,670 grams.
• Pumpkin K has a mass of 5,510,000 mg. To convert to grams, we need to divide by 1,000 or move the decimal point three places to the left. 5,510,000 mg \begin{align*}\rightarrow\end{align*} 5,510 grams.
• Pumpkin L has a mass of 567,100 cg. To convert to grams, we need to divide by 100 or move the decimal point two places to the left. 567,100 cg \begin{align*}\rightarrow\end{align*} 5,671 grams.
• Pumpkin M is already in grams \begin{align*}\rightarrow\end{align*} 5,800 grams.

The answer (from greatest to least) is Pumpkin M, Pumpkin L, Pumpkin J, Pumpkin K.

Now let’s use all of this information to solve the original problem.

## Real Life Example Completed

The Shot Put

Here is the original problem once again. Reread it and underline any important information.

When Marcus was in sixth grade he thought that he was a runner. So, he tried out for the track team and his love of running and determination made him an easy choice for the team. All that first year, Marcus ran. Then, in the hurdles, Marcus fell and hurt his knee. Marcus was devastated. He thought his life of track and field was over. While his knee healed, Marcus watched his teammates practice. One day after practice, Marcus picked up the shotput and threw it across the field. His coach was watching and Marcus found a new way to participate. “Sometimes life is like that,” his coach, Mr. Samuels, said. “You think you are going to be doing one thing and something else crosses your path.

This year, Marcus is on the shot put team. His team is trying to figure out which field they should use to practice in. There are several different fields available, but the team is large, so they are looking for a field with a width of 45.65 meters.

Field A—0.004565 kilometers

Field B—45,650 millimeters

Field C—456.5 centimeters

Field D—456,500 millimeters

Which field should his team choose?

First, we need to look at each unit of measurement to determine the best option for Marcus and his team. This problem asks us to compare the widths of these fields against 45.65 meters. Let’s begin by converting the widths of the each field into meters.

• Field A has a width of 0.004565 kilometers. To convert to meters, we multiply by 1,000 or move the decimal point three places to the right. 0.004565 kilometers=4.565 meters\begin{align*}0.004565 \ kilometers = 4.565 \ meters\end{align*}.
• Field B has a width of 45,650 millimeters. To convert to meters, we divide by 1,000 or move the decimal point three places to the left. 45,650÷1,000=45.65 meters\begin{align*}45,650 \div 1,000 = 45.65 \ meters\end{align*}.
• Field C has a width of 456.5 centimeters. To convert to meters, we divide by 100 or move the decimal point two places to the left. 456.5 centimeters÷100=4.565 meters\begin{align*}456.5 \ centimeters \div 100 = 4.565 \ meters\end{align*}.
• Field D has a width of 456,500 millimeters. To convert to meters, we divide by 1,000 or move the decimal point three places to the left. 456,500 millimeters=456.5 meters\begin{align*}456,500 \ millimeters = 456.5 \ meters\end{align*}.

Now we can compare the widths of the fields against 45.65 meters. Only Field B has the right width.

The solution is Field B.

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Metric System
a system of measurement developed by the French. Some units include meters, grams and liters.
Customary System
a system of measurement common in the United States. Some units include feet, pounds and gallons.
Estimate
to find an approximate measurement, useful in figuring out a reasonable number and not an exact one.
Equivalence
means equal.

## Technology Integration

Other Videos:

http://www.mathplayground.com/howto_Metric.html – This is a video that helps explain the metric system.

http://www.teachertube.com/members/viewVideo.php?title=Metric_conversion&video_id=169320 - This video provides a presentation of metric conversion.

## Time to Practice

Directions: Convert the following measurements into milliliters.

1. 65.57 liters

2. 28.203 centiliters

3. 0.009761 kiloliters

Directions: Convert the following measurements into centigrams.

4. 29.467 grams

5. 0.0562 milligrams

6. .0450584 kilograms

Directions: Convert the following measurements into kiloliters.

7. 89.96 liters

8. 45,217 milliliters

9. 3,120,700 centiliters

Directions: Compare or order the following measurements. Write <, >, or = for each ___.

10. 3.48 cl ___ 0.348 l

11. 57.21 kg ___ 572,100 cg

12. 91.17 mm ___ 0.09117 m

13. 4.4 cl ___ 0.44 ml

14. Order the following measurements from least to greatest: 79,282 kg, 7,838,200 cg, 7,938,200 mg, 79,382 g.

15. Order the following measurements from least to greatest: 2,261,000 cl, 21,061 l, 21.06 kl, 21,161,000 ml

Directions: Estimate a metric equivalent for the following customary measurements.

16. 28.5 ounces to grams

17. 11.75 inches to millimeters

18. 8 quarts to liters

19. 15 pounds to kilograms

Directions: Solve the following problems.

20. To join the swim team, Jessica swam every day. The distances she swam for the first four days were as follows: 1.65 km, 1,750 m, 185,000 cm, 1,950,000 mm. If the pattern continues, how many kilometers will she swim on the fifth day?

21. Mrs. Roth is moving to a new apartment. She can lift exactly 22.5 kg. Which of the following objects can she lift: box of books (23,500 g), statue (2,450,000 cg), computer (2,550,000 mg), potted plant (22.55 kg)?

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