# 3.3: Ordering Decimals

Difficulty Level: At Grade Created by: CK-12

## Introduction

Sizing Up Ice Cream Cones

So far Julie is really enjoying working at the ice cream stand. She loves talking with the people and the ice cream snacks are definitely a benefit.

However, she is very confused about the size of the ice cream cones.

Mr. Harris, the stand owner, used to be a math teacher so he loves to have fun with the customers. Because of this, the stand serves cones in different measurement units. It is famous for its mathematical ice cream cones.

This has been very frustrating for Julie.

Yesterday, a customer wanted to know whether a Kiddie Cone 1 was smaller or larger than a Kiddie Cone 2. One is in centimeters and one is in millimeters.

A second customer came in and wanted to know if the Small cone was larger than a Big Kid cone. Again, Julie didn’t know what to say.

Here is the chart of cone sizes.

Kiddie cone 1 = 80 mm

Kiddie cone 2 = 6 cm

Big Kid cone = 2.25 inches

Small cone = 2.5 inches

Julie went to see Mr. Harris for help, but he just chuckled.

“It is time to brush up on your measurement and decimals my dear,” he said smiling.

Julie is puzzled and frustrated.

Would you know what to say to the customers?

In this lesson, you will learn all about comparing. This lesson will teach you how to figure out which decimal or measurement is greater and which is smaller.

Hopefully, we will be able to help Julie at the end of the lesson.

What You Will Learn

In this lesson you will learn the following skills:

• Comparing Metric lengths
• Comparing decimals
• Ordering decimals
• Describing real-world portion or measurement situations by comparing and ordering decimals.

Teaching Time

I. Comparing Metric Length

In our last lesson we learned how to convert metric lengths. We learned that there are 10 millimeters in one centimeter and that we can change millimeters to centimeters by dividing. We also learned that we can change centimeters to millimeters by multiplying.

We can call these measurements equivalents.

The word equivalent means equals. When we know which measurement is equal to another measurement, then we can tell what is equal to what.

Here is a measurement chart of equivalents.

\begin{align*}1 \ cm & = 10 \ mm\\ 1 \ m & = 100 \ cm\\ 1 \ m & = 1000\ mm\\ 1 \ km & = 1000\ m\end{align*}

Let’s say that we wanted to compare two different units to figure out which is greater and which is less. We could use the chart to help us.

Here is an example.

Example

5 cm ______ 70 mm

1. The first thing that we need to do is to convert the measurements so that the unit of measurement is the same.

Here we have cm and mm. We need to have either all mm or all cm. It doesn’t matter which one we choose as long as it is the same unit. Let’s use cm.

\begin{align*}70 \ mm &= \underline{\;\;\;\;\;\;\;\;\;\;} \ cm\\ 70 \div 10 &= 7\end{align*}

2. Let’s rewrite the problem.

5 cm ______ 7 cm

3. Use greater than >, less than < or equal to = to compare the measurements.

Example

5 cm < 7 cm

So 5 cm < 70 mm

Take a minute to write down a few notes on these steps.

We can easily compare any two measurements once we have converted them to the same unit of measure.

Let’s look at another example

Example

7000 m ______ 8 km

Here we have two different units of measurement. We have meters and kilometers.

Our first step is to convert both to the same unit. Let’s convert to meters this time.

8 km \begin{align*}=\end{align*} 8 \begin{align*}\times\end{align*} 1000 \begin{align*}=\end{align*} 8000 m

Now we can compare.

7000 m < 8000 m

Our answer is 7000 m < 8 km.

Here are a few for you to try on your own. Use <, >, or = to compare.

1. 7 m ______ 7000 mm
2. 3 km ______ 3300 m
3. 1000 mm ______ 20 cm

Stop and check your work with a peer.

II. Compare Decimals

We just finished comparing metric lengths. All of the work that we did was with whole units of measurement. We compared which ones were greater than, less than or equal to. What if we had been working with decimals?

How can we compare decimals?

When we compare decimals, we are trying to figure out which part of a whole is greater. To do this, we need to think about the number one.

1 is a whole. All decimals are part of one.

The closer a decimal is to one, the larger the decimal is.

How can we figure out how close a decimal is to one?

This is a bit tricky, but if we look at the numbers and use place value we can figure it out.

Let’s look at an example.

Example

.45 ______ .67

Here we have two decimals that both have the same number of digits in them. It is easy to compare decimals that have the same number of digits in them.

Now we can look at the numbers without the decimal point. Is 45 or 67 greater?

67 is greater. We can say that sixty-seven hundredths is closer to one than forty-five hundredths.

This makes sense logically if you think about it.

Our answer is .45 < .67.

Steps for Comparing Decimals

1. If the decimals you are comparing have the same number of digits in them, think about the value of the number without the decimal point.
2. The larger the number, the closer it is to one.

What do we do if the decimals we are comparing don’t have the same number of digits?

Example

.567 ______ .64

Wow, this one can be confusing. Five hundred and sixty-seven thousandths seems greater. After all it is thousandths. The tricky part is that thousandths are smaller than hundredths.

Is this true?

To test this statement let’s look at a hundreds grid and a thousands grid.

Now it is easier to compare. You can see that .64 is larger than .567.

How can we compare without using a grid?

Sometimes, we don’t have a grid to look at, what then?

We can add zeros to make sure that digit numbers are equal. Then we can compare.

Let’s do that with the example we have been working on.

Example

.567 ______ .640

That made comparing very simple. 640 is larger than 567.

Our answer is that .567 < .640.

What about a decimal and a whole number?

Sometimes, a decimal will have a whole number with it. If the whole number is the same, we just use the decimal part to compare.

Example

3.4 ______ 3.56

First, we add in our zeros.

3.40 ______ 3.56

The whole number, 3 is the same, so we can look at the decimal.

40 is less than 56 so we can use our symbols to compare.

Our answer is 3.4 < 3.56.

Can you work these out on your own? Compare the following decimals using <, >, or =.

1. .0987 ______ .987
2. .453 ______ .045
3. .67 ______ .6700

How did you do? Take a minute to check your answers with a neighbor.

III. Order Decimals

Now that we know how to compare decimals, we can order them. Ordering means that we list a series of decimals according to size. We can write them from least to greatest or greatest to least.

How can we order decimals?

Ordering decimals involves comparing more than one decimal at a time. We need to compare them so that we can list them.

Here is an example for us to work with.

Example

.45, .32, .76

To write these decimals in order from least to greatest, we can start by comparing them.

The greater a decimal is the closer it is to one whole.

The smaller a decimal is the further it is from one whole.

Just like when we compared decimals previously, the first thing we need to look at is the digit number in each decimal. These each have two digits in them, so we can compare them right away.

Next, we can look at each number without the decimal and write them in order from the smallest to the greatest.

Example

.32, .45, .76

32 is smaller than 45, 45 is greater than 32 but smaller than 76, 76 is the largest number

Our answer is .32, .45, .76

What if we have decimals with different numbers of digits in them?

Example

Write these in order from greatest to least:

.45, .678, .23

Here we have two decimals with two digits and one decimal with three. We are going to need to create the same number of digits in all three decimals. We can do this by adding zeros.

Example

.450, .678, .230

Now we can write them in order from greatest to least.

Our answer is .23, .45, .678.

Now it is time for you to apply what you have learned. Write each series in order from least to greatest.

1. .6, .76, .12, .345
2. .34, .222, .6754, .5, .9
3. .78, .890, .234, .1234

Take a minute to check your work with a peer.

## Real Life Example Completed

Sizing Up Ice Cream Cones

Okay, now you have learned all about comparing measurement and decimals, so we can get back to Julie and the ice cream cones.

Let’s take another look at the problem first.

So far Julie is really enjoying working at the ice cream stand. She loves talking with the people and the ice cream snacks are definitely a benefit.

However, she is very confused about the size of the ice cream cones.

Mr. Harris, the stand owner, used to be a math teacher so he loves to have fun with the customers. Because of this, the stand serves cones in different measurement units. It is famous for its mathematical ice cream cones.

This has been very frustrating for Julie.

Yesterday, a customer wanted to know whether a Kiddie Cone 1 was smaller or larger than a Kiddie Cone 2. One is in centimeters and one is in millimeters.

A second customer came in and wanted to know if the Small cone was larger than a Big Kid cone. Again, Julie didn’t know what to say.

Here is the chart of cone sizes.

Kiddie Cone 1 = 80 mm

Kiddie Cone 2 = 6 cm

Big Kid cone = 2.25 inches

Small cone = 2.5 inches

Julie went to see Mr. Harris for help, but he just chuckled.

“It is time to brush up on your measurement and decimals my dear,” he said smiling.

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

Next, we can see that there are two customers who had questions.

Let’s look at the first customer’s question.

The first customer is comparing Kiddie Cone 1 with Kiddie Cone 2. Let’s look at the measurements for each of these cones.

Kiddie Cone 1 = 80 mm

Kiddie Cone 2 = 6 cm

We need to convert the units both to millimeters or both to centimeters.

Let’s use cm. We go from a smaller unit to a larger unit so we divide. There are 10 mm in 1 centimeters therefore we divide by 10.

80 \begin{align*}\div\end{align*} 10 = 8

Kiddie Cone 1 = 8 cm

Kiddie Cone 2 = 6 cm

8 > 6

Kiddie Cone 1 is greater than Kiddie Cone 2.

The second customer wanted to know whether the Big Kid Cone was larger or smaller than the Small cone.

These cones have measurements in decimals, so we need to compare the decimals.

Big Kid cone = 2.25

Small cone = 2.5

The whole number is the same, 2, so we can compare the decimal parts.

.25 and .50

.25 < .50

2.25 < 2.5

The Big Kid cone is smaller than the Small cone.

Julie is relieved. She now understands comparing decimals and measurement. Next time, she will be ready to answer any of the customer’s questions.

## Vocabulary

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

Equivalent
means equal
Comparing
using greater than, less than or equal to so that we can compare numbers
Decimals
a part of a whole represented by a number to the right of a decimal point
Order
writing numbers from least to greatest or greatest to least

## Technology Integration

Other Videos:

1. http://www.linkslearning.org/Kids/1_Math/2_Illustrated_Lessons/3_Place_Value/index.html – A GREAT video that starts with whole numbers and moves through to decimals. It really provides a clear understanding of the concepts.

## Time to Practice

Directions: Compare metric lengths using <, >, or =

1. 6 cm ______ 60 mm

2. 8 cm ______ 90 mm

3. 10 mm ______ 4 cm

4. 40 mm ______ 6 cm

5. 5 km ______ 4000 m

6. 7 km ______ 7500 m

7. 11 m ______ 1200 cm

8. 9 km ______ 9000 m

9. 100 mm ______ 750 cm

10. 18 km ______ 1500 m

Directions: Compare the following decimals using <, >, or =

11. .4 ______ .2

12. .67 ______ .75

13. .90 ______ .9

14. .234 ______ .54

15. .123 ______ .87

16. .954 ______ .876

17. .32 ______ .032

18. .8310 ______ .0009

19. .9876 ______ .0129

20. .8761 ______ .9992

Directions: Order the following decimals from least to greatest.

21. .8, .9. .2,. 4

22. .02, .03, .07, .05, .04

23. .34, .21, .05, .55

24. .07, .7, .007, .0007

25. .87, 1.0, .43, .032, .5

26. .067, .055, .023, .011, .042

27. .55, .22, .022, .033, .055

28. .327, .222, .0222, .321, .4

29. .65, .6, .67, .678, .69

30. .45, .045, 4.5, .0045, .00045

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