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# 3.4: Rounding Decimals

Difficulty Level: At Grade Created by: CK-12

## Introduction

The New Ice Cream Sign

Mr. Harris has given Jose the task of creating a new sign for “Add It Up Ice Cream”. The paint on the old sign is chipped and peeling, so Mr. Harris is hoping for a beautiful new sign to attract business.

Jose loves to paint and design things so he is the right person for the job. Jose is excited. He takes down the old sign and begins thinking about how he is going to design it.

Here is some of the information that Jose has to work with.

• The original sign is 4.25’ ×\begin{align*}\times\end{align*} 2.5’
• The letters on the original sign are 1.67' high

While Jose is working on his drawing, Mr. Harris walks up behind him.

“Jose, I think we should work with a new sign board too. Please round the length of the sign to the nearest half foot and the width to the nearest whole foot. Also, please make the letters a bit larger than the original. Maybe round up to the nearest foot on those too,” Mr. Harris says to Jose with a twinkle in his eye.

Jose smiles at Mr. Harris and then shrugs when Mr. Harris walks away.

Jose will need to remember how to round decimals for this plan to work.

In this lesson, you will need to learn how to round decimals to help Jose.

Pay close attention, we will be using what we learn in this lesson to help Jose with his new sign.

What You Will Learn

In this lesson you will learn the following skills:

• Round decimals using a number line.
• Round decimals given place value.
• Round very small decimal fractions to the leading digit
• Round very large numbers to decimal representations of thousands, millions, etc.

Teaching Time

Think about Jose. He is using decimals to design a new sign. His problem is an example of how decimals can show up in real life. Not all measurements are whole number measurements. Often we have measurements that are written in parts, decimals.

Sometimes, it is easier to round a decimal to the nearest whole or large part.

In this lesson, we are going to be learning how to round decimals.

I. Rounding Decimals Using a Number Line

Let’s think back for a minute to rounding whole numbers. When we were rounding whole numbers, we could round a number to any place value that we wanted to. We could round to tens, hundreds, thousands, etc.

To do this, we followed a few simple rules.

1. Look at the digit to the right of the place value you are rounding.
2. If the digit to the right is a five or greater, you round up.
3. If the digit to the right is less than 5, you round down.

Let’s look at an example to help us remember.

Example

Round the number 46 to the nearest ten

The four is in the tens place, that is the place we are rounding.

The six is in the ones place, that is the digit we look at.

Since 6 is a five or greater, we round up.

46 becomes 50.

There are a couple of different ways that we can round decimals.

First, let’s look at rounding them using a number line.

Here we have a number line. You can see that it starts with zero and ends with one. This number line has been divided up into quarters.

It goes from 0 to .25 to .50 to .75 to 1.0.

Let’s look at an example that we were going to round to the nearest quarter.

Example

.33

Here we have .33. The first thing that we want to do is to graph it on a number line.

We want to round to the nearest quarter. This number line gives us a terrific visual to do this.

Which quarter is .33 closest to?

It is closest to .25.

We can also round decimals to the nearest whole using a number line.

Example

Round 4.2 to the nearest whole number.

Here we can use our number line to show us which whole number 4.2 is closest too.

Wow! It is great to be able to see this so clearly.

Is 4.2 closer to 4.0 or 5.0 on the number line?

It is closer to 4.0.

II. Rounding Decimals to A Given Place Value

We can also use place value to help us in rounding numbers.

Once again, we are going to follow the same rules that we did when rounding whole numbers, except this time we will be rounding to the nearest whole or tens, hundreds, thousands, etc.

Let’s look at an example.

Example

Round .345 to the nearest tenth

To help us with this, let’s put the number in our place value chart.

Tens Ones Tenths Hundredths Thousandths

Ten

Thousandths

. 3 4 5

Now we are rounding to the nearest tenth.

3 is in the tenths place.

4 is the digit to the right of the place we are rounding.

It is less than 5, so we leave the 3 alone.

Notice that we don’t include the other digits because we are rounding to tenths. We could have put zeros in there, but it isn’t necessary.

Example

Round .567 to the nearest hundredth

To help us with this, let’s use our place value chart again.

Tens Ones Tenths Hundredths Thousandths

Ten

Thousandths

. 5 6 7

Now we are rounding to the nearest hundredth.

The 6 is in the hundredths place.

The 7 is the digit to the right of the hundredths place.

Since a 7 is 5 or greater, we round up to the next digit.

6 becomes 7.

Notice in this case that the five is included. Because it is to the left of the place we are rounding, it remains part of the number.

Now it’s time for you to practice, round each number using place value.

1. Round to the nearest tenth, .892
2. Round to the nearest hundredth, .632
3. Round to the nearest thousandths, .1238

Take a minute to go over your work with a neighbor.

III. Round Very Small Decimal Fractions to the Leading Digit

We know that a decimal is a part of a whole. We also know that some decimals are smaller than others. If we have a decimal that is 5 tenths of a whole, this is a larger decimal than 5 hundredths of a whole. Let’s look at those two decimals.

Example

.5 ______ .05

If we were going to compare these two decimals, we would add a zero to the first decimal so that it has the same number of digits as the second.

.50 > .05

We can see that the five tenths is greater than five hundredths.

This example can help us to determine very small decimals.

A decimal is a very small decimal depending on the number of places represented after the decimal point. The more decimal places, the smaller the decimal is.

Example

.000056787

Wow! That is a lot of digits. Because this decimal has so many digits, we can say that it is a very tiny decimal.

We can round tiny decimals like this one too. We use something called the leading digit to round a very small decimal.

The leading digit is the first digit of the decimal that is represented by a number not zero.

In this example, the leading digit is a five.

Example

.000056787

To round this decimal, we use the leading decimal and add in the rounding rules that we have already learned.

The digit to the right of the five is a six.

Six is greater than 5, so we round up.

Notice that we include the zeros to the left of the leading digit, but we don’t need to include any of the digits after the leading digit. That is because we rounded that digit so we only need to include the rounded part of the number.

We can find very small decimals in real life too. Look at this example.

Example

On August 5, 2007, the Japanese yen was worth .008467 compared to the US dollar.

Let’s say we wanted to round the worth of the yen to the leading digit.

First, let’s find the leading digit. The first digit represented by a number not a zero is 8.

Now we apply our rounding rules.

The digit to the right of the 8 is a 4. So the 8 remains the same.

It is your turn to apply this information, round each small decimal by using the leading digit.

1. .0004567
2. .0000178923
3. .00090034

Take a minute to check your work with a peer. Did you remember which value was the leading digit?

IV. Rounding Very Large Numbers to Decimal Representations of Thousands, Millions, etc.

We just finished rounding some very tiny numbers, but what about really large numbers? Can we use rounding to help us to examine some really large numbers?

Every time a new movie comes out a company keeps track of the total of the movie sales. If you go to www.the-numbers.com/movies/records you can see some of these numbers.

Here are the sales totals for the three top movies according to movie sales.

1. Star Wars IV - $460,998,007 2. Avatar -$558,179,737
3. Titanic - $600,788,188 Wow! Those are some big numbers! Here is where rounding can be very helpful. We can round each of these numbers to the nearest hundred million. First, let’s find the hundred millions place. 1. Star Wars IV -$460,998,007
2. Avatar - $558,179,737 3. Titanic -$600,788,188

We want to round to the nearest hundred million. We do this by looking at the number to the right of the place that we are rounding.

Let’s look at each movie individually.

1. Star Wars IV - The number after the 4 is a 6, so we round up to a 5. The rest of the numbers are zeros.

500,000,000

2. Avatar - The number after the 5 is a 5, so we round up to 6. The rest of the numbers are zeros.

600,000,000

3. Titanic - The number after the 6 is a zero. So the 6 stays the same and the rest of the numbers are zeros.

600,000,000

If we want to compare these numbers now we can see that Avatar and Titanic had the highest sales and Star Wars IV had the least sales.

Sometimes we can get confused reading numbers with so many digits in them. Rounding the numbers helps us to keep it all straight.

Here are a few for you to try. Round each to the correct place.

1. Round the nearest million, 5,689,432.
2. Round to the nearest hundred thousand, 789,345
3. Round to the nearest billion, 3,456,234,123

Take a minute to check your work with a peer.

## Real Life Example Completed

The New Ice Cream Stand

Now that you have had a chance to learn about rounding decimals, you are ready to help Jose with his dilemma.

Let’s look at the problem once again.

Mr. Harris has given Jose the task of creating a new sign for “Add It Up Ice Cream”. The paint on the old sign is chipped and peeling, so Mr. Harris is hoping for a beautiful new sign to attract business.

Jose loves to paint and design things so he is the right person for the job. Jose is excited. He takes down the old sign and begins thinking about how he is going to design it.

Here is some of the information that Jose has to work with.

• The original sign is 4.25’ ×\begin{align*}\times\end{align*} 2.5’
• The letters on the original sign are 1.67' high

While Jose is working on his drawing, Mr. Harris walks up behind him.

“Jose, I think we should work with a new sign board too. Please round the length of the sign to the nearest half foot and the width to the nearest whole foot. Also, please make the letters a bit larger than the original. Maybe round up to the nearest foot on those too,” Mr. Harris says to Jose with a twinkle in his eye.

Jose smiles and Mr. Harris and then shrugs when Mr. Harris walks away.

First, underline all of the important information. This has been done above.

There are two parts to Jose’s sign dilemma.

The first part is to round the length to the nearest half foot and the width of the original sign to the nearest foot.

Let’s look at the dimensions of the original sign: 4.25’ ×\begin{align*}\times\end{align*} 2.5’.

We want to round the length to the nearest half foot: 4.25 rounds to 4.5. Because the nearest half foot to .25 is .50.

The new length of the sign is 4.5’.

Next, we look at the width of the sign.

We want to round the width to the nearest foot, so we round 2.5’ to 3 feet.

The new width of the sign is 3 feet.

Jose has been having a trickier time with the sizing of the letters. The current size of the letters is 1.67’. He needs to round it to the nearest foot.

Let’s look at the decimal part of the measurement.

.67 is closer to one whole than to .50, so we round up.

This is actually quite simple. The question is whether 1.67 is closer to 1 or to 2. If we use the trick we have been practicing and look at the decimal along as if it were a whole number, then the question becomes: Is 67 closer to 0 or to 100? Since 67 is obviously closer to 100, .67 is closer to 1. Since we have already 1 whole, we add 1 more whole, and as a result, 1.67 feet rounds to 2 feet.

You can use the rules for rounding whenever you are rounding any decimal.

## Vocabulary

Here are the vocabulary words that you will find throughout this lesson.

Round
to use place value to change a number whether it is less than or greater than the digit in the number
Decimal
a part of a whole written to the right of a decimal point. The place value of decimals is marked by THS (such as tenTHS, hundredTHS, etc).
the first digit of a tiny decimal that is not a zero
Small decimals
decimals that have several zeros to the right of the decimal point before reaching a number.

## Technology Integration

Other Videos:

This video shows two students in the sixth grade explaining how to round decimals.

## Time to Practice

Directions: Use the number line and round to the nearest decimal on the number line.

1. 2.54

2. 2.12

3. 2.78

4. 2.89

5. 2.33

6. 2.42

7. 2.97

8. 2.01

9. 2.11

10. 2.27

Directions: Round according to place value

11. Round .45 to the nearest tenth

12. Round .67 to the nearest tenth

13. Round .123 to the nearest tenth

14. Round .235 to the nearest hundredth

15. Round .567 to the nearest hundredth

16. Round .653 to the nearest hundredth

17. Round .2356 to the nearest thousandth

18. Round .5672 to the nearest thousandth

19. Round .8979 to the nearest thousandth

20. Round .1263 to the nearest thousandth

Directions: Round each to the leading digit.

21. .0045

22. .0067

23. .000546

24. .000231

25. .000678

26. .000025

27. .000039

28. .000054

29. .0000278

30. .0000549

Directions: Round each number to the specified place value.

31. 5,689,123 to the nearest million

32. 456,234 to the nearest ten thousand

33. 678,123 to the nearest thousand

34. 432,234 to the nearest hundred thousand

35. 567,900 to the nearest thousand

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