# 3.1: Decimal Place Value

Difficulty Level: At Grade Created by: CK-12

## 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.

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.

1. 567
2. .345
3. .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

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

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

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.

1. .7
2. .765
3. .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.

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.

Other Videos:

1. 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

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