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 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 Concept is all about. It is perfect timing for Karina too. At the end of the Concept, you will be able to help her with her homework.**

### Guidance

**Previously we worked 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 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. Remember 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*}

**Let's look at how we can use this information.**

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.**

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 problem.**

\begin{align*}23 \times 1000 & = 23,000\\ 23 \ kilograms &= 23,000 \ grams.\end{align*}

Convert the following units to their equivalent.

#### Example A

10 centigrams = _____ grams

**Solution: \begin{align*}0.1\end{align*} grams**

#### Example B

1 meter = _____ millimeters

**Solution: \begin{align*}1000\end{align*} millimeters**

#### Example C

15 kilometers = _____ meters

**Solution: \begin{align*}15,000\end{align*} meters.**

Now back to the javelin dimensions.

Reread the original problem. 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 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 with 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 original 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!**

### Guided Practice

Here is one for you to try on your own.

Convert 25,000 meters into kilometers.

**Answer**

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*}

**This is our answer.**

### Video Review

This is a James Sousa video on metric system conversions and equivalence.

### Explore More

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

15. How many liters in one kiloliter?