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This lesson submitted by jmdavidson01

*What You Will Learn*

In this lesson, you will learn the following skills:

- Identify the equivalence of metric units of length
- Measure lengths using metric units to the nearest decimal place.
- Choose appropriate tools for given decimal metric measurement situations
- Choose appropriate decimal units for given metric measurement situations

### Vocabulary

Here are the vocabulary words that can be found in this unit.

- Metric System
- a system of measurement more commonly used outside of the United States

- Length
- the measurement of a object or distance from one end to the other

- Millimeter
- the smallest common metric unit of measuring length, found on a ruler

- Centimeter
- a small unit of measuring length, found on a ruler

- Meter
- approximately 3 feet, measured using a meter stick

- Kilometer
- a measurement used to measure longer distances, the largest common metric unit of measuring length

## Decimal Place Value

### Introduction

*The Ice Cream Stand*

Julie and her friend Jose are working at an ice cream stand for the summer. They are excited because in addition to making some money for the summer, they also get to eat an ice cream cone every day.

On the first day on the job, Julie is handed a cash register drawer that is filled with money. This is the drawer that she can collect money from sales in as well as make change for customers.

Julie needs to count the amount of money in her drawer to be sure that it is accurate. Her boss Mr. Maguire tells her that her drawer should have ** sixty-five dollars and seventy-five cents** in it.

He hands her a data sheet that she needs to write that money amount in on.

Julie looks at the bills in her drawer and begins to count. She finds 2-20 dollar bills, 2-ten dollar bills, 1-five dollar bill and 2 quarters, 2 dimes and 1 nickel.

**Now it is your turn to help.**

**In this lesson, you will learn all about decimals. One of the most common places that we see decimals is when we are working with money. Your work with decimals and place value will help Julie count her bills and change accurately.**

**Pay attention so that you can count and write the correct amount of money on Julie’s data sheet at the end of the lesson.**

*What You Will Learn*

In this lesson, you will learn how to complete the following tasks:

- Express numbers given in words or hundredths grids using decimal place value.
- Express numbers in expanded form given decimal form.
- Read and write decimals to ten-thousandths place.
- Write combinations of coins and bills as decimal money amounts.

*Teaching Time*

I. **Express Numbers Given in Words or Hundredths Grids Using Decimal Place Value**

Up until this time in mathematics, we have been working mainly with ** whole numbers**. A whole number represents a whole quantity. There aren’t any parts when we work with a whole number.

When we have a part of a whole, we can write it in a couple of different ways. One of the ways that we write it is as a ** decimal**.

**A decimal is a part of a whole**. Here is an example of a decimal.

Example

4.56

This decimal has parts and wholes in it. Notice that there is a point in the middle of the number. This is called the ** decimal point**. The decimal point helps us to divide the number between wholes and parts.

**To the right of the decimal point are the parts of the whole and to the left of the decimal point is the whole number.**

We can have numbers with parts and wholes in them, and we can have numbers that are just decimals.

Example

.43

This decimal has two decimal places. Each digit after the decimal is in a different place. We call these places place values.

When you were working with whole numbers you used place value too, but this is a new place value system that includes decimals.

**How can we express a decimal using place value?**

To express a decimal using place value we need to use a place value chart. This gives us an idea about the worth of the decimal.

Here is a place value chart.

**Tens**

**Ones**

**Tenths**

**Hundredths**

**Thousandths**

**Ten**

**Thousandths**

**.**

Notice that if we take the last example and write it in the place value chart above each number is a word. That word gives us the value of that digit according to its place in the chart. This number is forty-three hundredths. The three is the last number, and is in the hundredths place so that lets us know to read the entire number as hundredths.

**Tens**

**Ones**

**Tenths**

**Hundredths**

**Thousandths**

**Ten**

**Thousandths**

**.**

**4**

**3**

Hmmm. Think about that, the word above each digit has a name with a THS in it. The THS lets us know that we are working with a part of a whole.

**What whole is this decimal a part of?**

To better understand what whole the decimal is a part of, we can use a picture. We call these grids or hundreds grids. Notice that the number in the last example was .43 or 43 hundredths. The hundredths lets us know that this is “out of one hundred.”

Here is a picture of a hundreds grid.

Now we want to show 43 hundredths of the hundreds grid. To do that, we shade 43 squares. Each square is one part of one hundred.

**What about tenths?**

If you look at a place value chart, you can see that there are other decimal names besides hundredths. We can also have tenths.

Example

.5

Here is a number that is five-tenths. We can create a picture of five-tenths using a grid of ten units.

If we want to show .5 in this box, we can see that tenths means 5 out of 10. We shade five boxes of the ten.

We can make pictures of tenths, hundredths, thousandths and ten-thousands.

Ten-thousandths, whew! Think about how tiny those boxes would be.

**Here are a few for you to try. Write each number in words and as a decimal using each grid.**

*Take a minute to check your work with a peer.*

II. **Express Numbers in Expanded Form Given Decimal Form**

We just worked on expressing decimals in words using a place value chart and in pictures using grids with tens and hundreds in them.

We can also stretch out a decimal to really see how much value each digit of the decimal is worth. This is called *expanded form.*

**What is expanded form?**

Expanded form is when a number is stretched out. Let’s look at a whole number first and then use this information with decimals.

Example

265

If we read this number we can read it as two hundred and sixty-five.

**We can break this apart to say that we have two hundreds, six tens and five ones.**

**HUH??? What does that mean? Let’s look at our place value chart to help us make sense of it.**

**Hundred**

**Tens**

**Ones**

**Tenths**

**Hundredths**

**Thousandths**

**Ten**

**Thousandths**

**2**

**6**

**5**

**.**

If you look at the chart you can see how we got those values for each digit. The two is in the hundreds place. The six is in the tens place and the five is in the ones place.

Here it is in expanded form.

2 hundreds + 6 tens + 5 ones

**This uses words, how can we write this as a number?**

200 + 60 + 5

Think about this, two hundred is easy to understand. Six tens is sixty because six times 10 is sixty. Five ones are just that, five ones.

**This is our number in expanded form.**

**How can we write decimals in expanded form?**

We can work on decimals in expanded form in the same way. First, we look at a decimal and put it into a place value chart to learn the value of each digit.

Example

.483

**Hundred**

**Tens**

**Ones**

**Tenths**

**Hundredths**

**Thousandths**

**Ten**

**Thousandths**

**.**

**4**

**8**

**3**

Now we can see the value of each digit.

4 = four tenths

8 = eight hundredths

3 = 3 thousandths

We have the values in words, now we need to write them as numbers.

Four tenths = .4

Eight hundredths = .08

Three thousandths = .003

**What are the zeros doing in there when they aren’t in the original number?**

The zeros are needed to help us mark each place. We are writing a number the long way, so we need the zeros to make sure that the digit has the correct value.

If we didn’t put the zeros in there, then .8 would be 8 tenths rather than 8 hundredths.

Now, we can write this out in expanded form.

Example

.483

**.4 + .08 + .003 = .483**

**This is our answer in expanded form.**

**Now it is your turn. Write each number in expanded form.**

**567****.345****.67**

*Check your work with a friend to be sure that you are on the right track.*

III. **Read and Write Decimals to the Ten-Thousandths Place**

We have been learning all about figuring out the value of different decimals. We have used place value to write them, we have used pictures and we have stretched them out. Now it is time to learn to read and write them directly. Let’s start with reading decimals.

**How do we read a decimal?**

We read a decimal by using the words that show the place value of the last digit of the decimal. That may sound confusing, so let’s look at an example.

Example

.45

To help us read this decimal, we can put it into our place value chart.

**Hundred**

**Tens**

**Ones**

**Tenths**

**Hundredths**

**Thousandths**

**Ten**

**Thousandths**

**.**

**4**

**5**

We read this decimal by using the place value of the last digit to the right of the decimal point.

Normally, we would read this number as **forty-five.**

Because it is a decimal, we read forty-five hundredths. The last digit is a five and it is in the hundredths place.

**Can we use place value to write the number too?**

Yes we can. We write the number as we normally would.

Example

Forty-five

Next, we add the place value of the last digit to the right of the decimal point.

Forty-five hundredths

**Our answer is forty-five hundredths.**

**We can use this method to read and write any decimal. What about a decimal with more digits?**

Example

.5421

First, let’s put this number in our place value chart.

**Hundred**

**Tens**

**Ones**

**Tenths**

**Hundredths**

**Thousandths**

**Ten**

**Thousandths**

**.**

**5**

**4**

**2**

**1**

**First, let’s read the number.**

We can look at the number without the decimal. It would read:

**Five thousand four hundred twenty-one**

Next we add the place value of the last digit

**Ten thousandth**

**Five thousand four hundred and twenty-one ten thousandths**

**This is our answer.**

**It is also the way we write the number in words too. Notice that is it very important that we add the THS to the end of the place value when working with decimals.**

**Alright, now you try a few. Write each decimal in words.**

**.7****.765****.2219**

*Take a minute to check your work with a peer.*

IV. **Write Combinations of Coins and Bills as Decimal Money Amounts**

**How can we apply what we have learned in a real world way?**

**Money** is a way that we use decimals every day. Let’s think about change.

Coins are cents. If we have 50 pennies, then we have 50 cents. It takes 100 pennies to make one dollar or one whole.

Coins are parts of one dollar. We can represent coins in decimals.

**Let’s start with pennies.**

A penny is one cent or it is one out of 100.

When we have a collection of pennies, we have so many cents out of 100.

Example

5 pennies is 5 cents.

**How can we write 5 cents as a decimal?**

To do this, we need to think about 5 out of 100.

**We can say that 5 cents is 5 hundredths of a dollar since there are 100 pennies in one dollar.**

**Let’s write 5 cents as a decimal using place value.**

**Hundred**

**Tens**

**Ones**

**Tenths**

**Hundredths**

**Thousandths**

**Ten**

**Thousandths**

**.**

**5**

The five is in the hundredths box because five cents is five one hundredths of a dollar.

We need to add a zero in the tenths box to fill the gap.

**Hundred**

**Tens**

**Ones**

**Tenths**

**Hundredths**

**Thousandths**

**Ten**

**Thousandths**

**.**

**0**

**5**

**Now we have converted 5 cents to a decimal.**

**How can we write 75 cents as a decimal?**

First, think about what part of a dollar 75 cents is.

Seventy-five cents is seventy-five out of 100.

Now, we can put this into our place value chart.

**Hundred**

**Tens**

**Ones**

**Tenths**

**Hundredths**

**Thousandths**

**Ten**

**Thousandths**

**.**

**7**

**5**

**Now we have written it as a decimal.**

**What about when we have dollars and cents? Suppose we have twelve dollars and fourteen cents.**

**A dollar is a whole number amount. Dollars are found to the left of the decimal point.**

**Cents are parts of a dollar. They are found to the right of the decimal point.**

**How much money do we have?**

**There is one ten and the two ones gives us twelve dollars.**

**Then we have some change. One dime and four pennies is equal to fourteen cents.**

Here are the numbers:

12 wholes

14 parts

**Let’s put them in our place value chart.**

**Hundred**

**Tens**

**Ones**

**Tenths**

**Hundredths**

**Thousandths**

**Ten**

**Thousandths**

**1**

**2**

**.**

**1**

**4**

**There is our money amount.**

**Our answer is $12.14.**

*Notice that we added a dollar sign into the answer to let everyone know that we are talking about money.*

### Real Life Example Completed

*The Ice Cream Stand*

**Now that we know about decimals and money we are ready to help Julie with her ice cream shop dilemma.**

Julie and her friend Jose are working at an ice cream stand for the summer. They are excited because in addition to making some money for the summer, they also get to eat an ice cream cone every day.

On the first day on the job, Julie is handed a cash register drawer that is filled with money. This is the drawer that she can collect money from sales in as well as make change for customers.

Julie needs to count the amount of money in her drawer to be sure that it is accurate. Her boss Mr. Maguire tells her that her drawer should have ** sixty-five dollars and seventy-five cents** in it.

He hands her a data sheet that she needs to write that money amount in on.

Julie looks at the bills in her drawer and begins to count. She finds 2-20 dollar bills, 2-ten dollar bills, 1-five dollar bill and 2 quarters, 2 dimes and 1 nickel.

**First, let’s underline all of the important information.**

**Now, let’s count the money she has in the drawer.**

1. **How many whole dollars are there?**

There are 2 Twenty Dollar bills = $40 plus 2 Ten Dollar bills = $20 plus 1 Five Dollar bill = $5.

The total then is $40 + $20 + $5 = $65.

2. **How many cents are there?**

There are 2 Quarters at $25. each = $.50 plus 2 Dimes at $.10 each = $.20 plus 1 Nickel at $.05 = $.05

The total then is $.50 + $.20 + $.05 = $.75

**Our next step is to write the wholes and parts in the place value chart. Then we will have this written as a money amount.**

**Hundred**

**Tens**

**Ones**

**Tenths**

**Hundredths**

**Thousandths**

**Ten**

**Thousandths**

**6**

**5**

**.**

**7**

**5**

**Great work!! Julie has $65.75 in her drawer. That is the correct amount. She is ready to get to work.**

### Vocabulary

Here are the vocabulary words that can be found in ** italics** throughout the lesson.

- Whole number
- a number that represents a whole quantity

- Decimal
- a part of a whole

- Decimal point
- the point in a decimal that divides parts and wholes

- Expanded form
- writing out a decimal the long way to represent the value of each place value in a number

### Technology Integration

This video presents an example of expanded place value.

Khan Academy Decimal Place Value

James Sousa, Write a Number in Decimal Notation from Words

Other Videos:

- http://www.teachertube.com/viewVideo.php?title=Money_Fractions_and_Decimals&video_id=59116&vpkey=4badb7d45d – This video is a short story and features two students learning about money with fractions and decimals.

### Time to Practice

Directions: Look at each hundreds grid and write a decimal to represent the shaded portion of the grid.

1.

2.

3.

4.

5.

Directions: Write each decimal out in expanded form.

6. .78

7. .345

8. .98

9. .231

10. .986

11. .33

12. .821

13. .4321

14. .8739

15. .9327

Directions: Write out each decimal in words.

16. .4

17. .56

18. .93

19. .8

20. .834

21. .355

22. .15

23. .6

24. .5623

25. .9783

## Measuring Metric Length

### Introduction

*The Kid’s Area*

There are a lot of children who visit the ice cream stand each week. Most times they sit with their parents at a large picnic table.

Jose has collected a few small picnic tables to put near each other for a small “kid’s area.” Mr. Harris loves the idea. Jose gets to work arranging the tables.

Jose has four small picnic tables for his kid’s area. He wants to put the tables about 1.5 meters apart. He thinks that this will give the kids plenty of room to not be on top of each other.

He puts out the tables and then gets a ruler and a meter stick. Which tool should Jose use to measure the distance between the two tables?

If he wants the tables to be 1.5 meters apart, how many meter sticks will the distance actually be?

Once Jose gets the tables set up, he wants to design a new placemat for the kids to eat off of. For his placemat, should Jose use a ruler or a meter stick when he measures out the design?

Which makes more sense?

**This lesson is all about metric measurement. In the end of the lesson, you will be able to help Jose with his kid’s area.**

**Pay close attention! In the United States we don’t have a lot of experience with Metrics. You will need all of the information in this lesson to be successful.**

I. **Identify Equivalence of Metric Units of Length**

This lesson focuses on metric units of measurements. In the United States, we use the customary system of measurement more than we use the ** metric system** of measurement. However, if you travel to another country or complete work in science class, you will need to know metrics.

**What are the metric units for measuring length?**

When measuring ** length**, we are measuring how long something is, or you could say we are measuring from one end to the other end. That is the length of the item.

**Here are the common metric units of length from the smallest unit to the largest unit.**

*Millimeter*

*Centimeter*

*Meter*

*Kilometer*

A ** millimeter** is the smallest unit. Millimeters are most useful when measuring really tiny things. You can find millimeters on some rulers.

A ** centimeter** is the next smallest unit. Centimeters can also be found on a ruler.

A ** meter** is a little more than 3 feet. A meter is a unit that would be very helpful to a carpenter or to someone working in construction.

A ** kilometer** is used to measure longer distances. You often hear the word kilometer mentioned when talking about a road race that is 5k (or 5 kilometers) long.

**How can we convert metric units of length?**

When working with the customary units of length: inches, feet, etc., we know that we can convert them from one to another to change the units we are working with. For example, if you have 24 inches, it might make more sense to say that we have 2 feet.

We can do the same thing when working with metric units.

Here is a chart to help us with the conversions.

& 1 \ m \qquad \ \; 100 \ cm\\

& 1 \ cm \qquad \; 10 \ mm" />

Now that you know the conversions, we can change one unit to another unit. Let’s look at an example.

Example

5 km = ____ m

Here we are converting kilometers to meters.

**How can we convert larger units to smaller units?**

We can convert larger units to smaller units by multiplying.

There are 1000 meters in one kilometer.

Example

5 \times 1000 &= 5000 \ m" />

**Our answer is 5000 m.**

Example

600 cm = ______ m

Here we are converting smaller units to larger units.

**How can we convert smaller units to larger units?**

We can convert smaller units to larger units by dividing.

There are 100 cm in one meter.

Example

600 100 6

**Our answer is 6.**

**Now it’s time for you to try some. Complete the following conversions.**

**2000 mm = ______ cm****3 km = ______ m****4000 cm = ______ m**

II. **Measure Lengths Using Metric Units to the Nearest Decimal Place**

Sometimes when we convert metric units we don’t have a whole number answer. In the last section, all of the examples ended with whole numbers.

Example

2000 mm = 200 cm

These are both whole numbers.

**What happens when we convert smaller units to larger units and they don’t end up as a whole number?**

When this happens, we end up with an answer that is a decimal. If we remember our rules for working with decimals and place value, we can be very successful at converting these small units of measurement to larger units.

Example

1 mm = ______ cm

Here we are converting a smaller unit to a larger unit, because of this we know that we are going to divide.

There are 10 mm in one centimeter, so we are going to divide 1 by 10.

Think about this, we are dividing 1 whole into 10 parts-our answer is definitely going to be a decimal.

1 10 .1 (one tenth)

**Our answer is that 1 mm = .1 cm.**

**We can also round our answer to the nearest tenth.**

What if we had a problem where we wanted to convert 1.5 mm to cm?

Example

1.5 mm = ______ cm

Once again, we are going to be dividing by 10.

When we divide by 10 in this example we end up with an answer of .15

**1.5 mm = .15 cm**

**We can round this answer to the nearest tenth.**

**.15 rounds to .2**

**We can say that .2 is the closest tenth of a cm to 1.5 mm.**

**Just as we were able to round whole numbers, we can round decimal measurements too.**

Let’s look at another example where we will get a decimal answer.

Example

1 m = ______ km

Here we are converting a smaller unit to a larger unit.

There are 1000 meters in one kilometer. We divide by 1000.

**1 1000 .001**

**Here our answer is one-thousandth of a kilometer.**

**Now it is time for you to try a few.**

**2 m = ______ km****8 mm = ______ cm****4 cm = ______ m**

III. **Choose Appropriate Tools Given Decimal Metric Measurement Situations**

Now that you have learned all about converting different measurements, it is time to think about which tools to use to measure different things.

We know some metric units for measuring length are millimeters, centimeters, meters and kilometers.

Millimeters and centimeters are found on a ruler.

There is a meter stick that measures 1 meter.

A metric tape measure can be used to measure multiple meters.

If you wanted to measure long distances, you could use a kilometer odometer, like in a car, to measure distance.

**What tool should we use when?**

A tool is designed to make measuring simpler. If we have a difficult time choosing an appropriate tool, or choose a tool that isn’t the best choice, it can make measuring very challenging.

Let’s think about tools and when we should use them depending on what and/or where we are measuring.

**Here are some general suggestions:**

**If the object is very tiny, use a ruler for millimeters.** **If the object is less than 30 cm use a ruler for centimeters.** **If the object is between 30 cm and 5 or so meters use a meter stick.** **If the object is greater than a few meters, use a metric tape measure.** **If the object is a long distance, for instance across town, use a kilometer odometer.**

Example

What would we use to measure the following object?

This object is a paperclip. It is definitely smaller than the length of a ruler, so we can use a ruler to measure it.

Example

What about measuring a road race?

A road race is usually a significant distance, so we are going to use a kilometer odometer to measure it.

**Now it is time for you to choose an appropriate tool.**

- The width of a table
- An ant

IV. **Choose Appropriate Decimal Units for Given Metric Measurement Situations**

Now that we know about using the appropriate tool, we also need to choose the best unit to measure different things.

The common metric units of length are *millimeter, centimeter, meter and kilometer.*

**When is the best time to use each measurement?**

You can think about this logically. Let’s start with millimeters.

**A millimeter is the smallest unit. There are 10 mm in one centimeter, if an object is smaller than one centimeter, then you would use millimeters.**

**Who would use millimeters?** A scientist measuring something under a magnifying glass might use millimeters to represent a tiny specimen.

**A centimeter is the next smallest unit. We can use a ruler to measure things in centimeters. If an object is the length of a ruler or smaller, then it makes sense to use centimeters to measure.**

**Meters are used to measure everything between the length of a ruler and the distance between two cities or places.**

Most household objects such as tables, rooms, window frames, television screens, etc would be measured in meters.

**Kilometers are used to measure distances. If we are looking to figure out the length of a road, the distance between two locations, etc, we would use kilometers.**

**Think about each example, which is the best unit to measure the objects listed below?**

**The height of a picture on the wall****A caterpillar**

*Explain your answers to a neighbor. Be sure to justify why you chose each unit of measurement.*

### Real Life Example Completed

*The Kid’s Area*

**Now that you have worked with the Metric System, let’s go back and look at Jose’s work with the kid’s area.**

**Here is the problem once again.**

There are a lot of children who visit the ice cream stand each week. Most times they sit with their parents at a large picnic table.

Jose has collected a few small picnic tables to put near each other for a small “kid’s area.” Mr. Harris loves the idea. Jose gets to work arranging the tables.

Jose has four small picnic tables for his kid’s area. He wants to put the tables about 1.5 meters apart. He thinks that this will give the kids plenty of room to not be on top of each other.

He puts out the tables and then gets a ruler and a meter stick. Which tool should Jose use to measure the distance between the two tables?

If he wants the tables to be 1.5 meters apart, how many meter sticks will the distance actually be?

Once Jose gets the tables set up, he wants to design a new placemat for the kids to eat off of. For his placemat, should Jose use a ruler or a meter stick when he measures out the design?

Which makes more sense?

**First, let’s underline the important questions and information in this problem.**

**Now let’s look at the first question. Jose wants to measure a distance that is much longer than a ruler. He could use a ruler, but think about how many centimeters are in one meter. If Jose is wishing to make his work the simplest that it can be, then he should use the meter stick.**

**For 1.5 meters, Jose would have to measure out 150 centimeters.**

**If Jose uses the meter stick, then he would need to measure one and one-half lengths of the meter stick to have the accurate measurement between the tables.**

**For the placemat design, Jose is going to be working with a much smaller area. He can use a ruler for this design since most placemats are about the size of a piece of paper. Jose will be able to work well with his ruler while a meter stick would be very difficult to work with.**

Homework:

Directions: Complete the following metric conversions.

1. 6 km = ______ m

2. 5 m = ______ cm

3. 100 cm = ______ m

4. 400 cm = ______ m

5. 9 km = ______ m

6. 2000 m = ______ km

7. 20 mm = ______ cm

8. 8 cm = ______ mm

9. 900 cm = ______ m

10. 12 m = ______ cm

Directions: Write each decimal conversion. Round to the nearest hundredth when necessary

11. 1 mm = ______ cm

12. 5 mm = ______ cm

13. 8 cm = ______ m

14. 9 cm = ______ m

15. 12 m = ______ km

16. 8 m = ______ km

17. 22 mm = ______ cm

18. 225 mm = ______ cm

19. 543 mm = ______ cm

20. 987 mm = ______ cm

Directions: Choose the best tool to measure each item. Use ruler, meter stick, metric tape or kilometric odometer.

21. A paperclip

22. The width of a dime

23. A tall floor lamp

24. The width of a room

25. A road race from start to finish

Directions: Choose the best metric unit for each measurement situation.

26. The length of a small table

27. A book

28. A cell phone

29. The length of a room

30. The distance from Boston to Cincinnati