4.3: Multiplying Decimals
Introduction
The Triceratops Skeleton
When the students in Mrs. Andersen’s class arrive at the Science Museum, Kara is very excited to learn that there is a dinosaur exhibit. In fact, it is a famous dinosaur exhibit. A set of dinosaur bones from a triceratops has been reconstructed and is on display.
Kara can’t wait to get to see it. She has a feeling that this is going to be her favorite part of the museum. Several other students are equally excited, so Mrs. Andersen and the chaperones decide to go to the exhibit first and the split up into groups.
When Kara walks in, she is delighted. There before her eyes is a huge skeleton of a triceratops. After visiting the exhibit for a while, the students begin to move on. Mrs. Andersen sees Kara hesitate before leaving the exhibit. She walks over to her.
“Imagine, that dinosaur is about 4 and a half times as long as you are!” Mrs. Andersen smiles.
The students exit the exhibit hall, but Kara pauses at the door. She has to think about this. In all of her excitement she forgot to find the information that actually says how long the triceratops actually is.
Mrs. Andersen’s words stay with her, “the dinosaur is \begin{align*}4 \frac{1}{2}\end{align*} times as long as you are.”
Kara knows that she is \begin{align*}5 \frac{1}{4}\end{align*} feet tall. If the dinosaur is \begin{align*}4 \frac{1}{2}\end{align*} times as long as she is, how long is the dinosaur?
While Mrs. Andersen and the chaperones start to split up the students, Kara begins working some quick math on the back of her museum map.
She writes down the following figures.
5.25 \begin{align*}\times\end{align*} 4.5 \begin{align*}=\end{align*} ______
If Kara multiplies these numbers correctly, she will be able to figure out how long the triceratops is.
How long is he?
In this lesson you will learn all about multiplying decimals. When finished, you will know the length of the triceratops.
What You Will Learn
In this lesson you will learn the following skills:
- Multiply decimals by decimals using area models (hundredths grid).
- Place the decimal point in the product and confirm by estimation.
- Multiply decimals up to a given thousandths place.
- Solve real-world problems involving area of rectangles with decimal dimensions.
Teaching Time
I. Multiply Decimals by Decimals Using Area Models (hundredths grid)
Sometimes in life, you will need to multiply a decimal by another decimal. In our last lesson, you learned to multiply a decimal and a whole number. In this lesson, you will learn how to multiply a decimal with another decimal.
Let’s start by thinking of a decimal in terms of a picture. We can use a hundreds grid to represent the hundredths of a decimal.
0.3 = 0.30 = 30 hundredths
Shade 30 squares green because we are looking at 30 out of 100 or 30 hundredths.
Let’s say that that is our first decimal. We are going to multiply it with another decimal. Let’s say that we are going to multiply .30 \begin{align*}\times\end{align*} .40.
Here is a visual picture of what .40 or 40 hundredths looks like.
0.4 = 0.40 = 40 hundredths
Shade 40 squares yellow.
Now we have two visuals of the decimals that we are multiplying. If we put them both together, then we can see what it would look like to multiply these two decimals together.
Notice that the overlapping part is the product of this problem.
Our answer is .12 or 12 hundredths.
II. Place Decimal Point in the Product and Confirm by Estimation
Drawing a couple of hundreds grids each time you wish to multiply isn’t really a practical way to go about multiplying.
How can we multiply two decimals without using a hundreds grid?
One of the ways that we can do it is to work on it just like we did when we multiplied decimals and whole numbers together.
First, we ignored the decimal point and multiplied just like it was two whole numbers that we were multiplying.
Second, we counted our decimal places and inserted the decimal into the product when we had finished multiplying.
We can approach two decimal multiplication in the same way.
Let’s look at an example.
Example
1.3 \begin{align*}\times\end{align*} .24 = ______
To work on this problem, let’s start by writing it vertically instead of horizontally. Then we multiply.
Example
\begin{align*}1.3 \\ \underline{\times \quad .24} \\ 52 \\ \underline{+ \ \ 260} \\ 312\end{align*}
Now that we have finished the other steps, our final step is to put the decimal point in the correct spot.
To do this, we need to count the decimal places in each number from right to left. The first number has one decimal place.
1.3
The second number has two decimal places.
.24
This is a total of three decimal places that need to be placed into the product.
Our final answer is .312.
How can we confirm our answer by using estimation?
When we multiply two decimal, sometimes we can use estimation to check our work.
Let’s look at an example.
Example
4.7 \begin{align*}\times\end{align*} 2.1 \begin{align*}=\end{align*} ______
We can start by rounding each decimal to the nearest whole number.
4.7 rounds to 5.
2.1 rounds to 2.
Next, we multiply 5 \begin{align*}\times\end{align*} 2 \begin{align*}=\end{align*} 10.
Our answer is around 10.
Now let’s figure out our actual answer and see if our estimate is reasonable.
Example
\begin{align*}4.7 \\ \underline{\times \quad \ 2.1} \\ 47 \\ \underline{+ \quad 940} \\ 9.87\end{align*}
Our answer is 9.87.
We can see that our estimate is reasonable because 9.87 is very close to 10.
Now it is your turn. Write an estimate for each example and then multiply for the actual answer.
- 3.1 \begin{align*}\times\end{align*} 4.9 \begin{align*}=\end{align*} _____
- 1.2 \begin{align*}\times\end{align*} 5.1 \begin{align*}=\end{align*} _____
- 3.2 \begin{align*}\times\end{align*} 6.7 \begin{align*}=\end{align*} _____
Take a minute to check your work with a peer. Is your estimate reasonable? Is your multiplication accurate?
III. Multiply Decimals Up to a Given Thousandths Place
We can use what we have learned to multiply decimals that have many more places too. These are small decimals. Remember that the greater the number of decimal places after the decimal point, the smaller the decimal actually is.
Let’s look at an example.
Example
.134 \begin{align*}\times\end{align*} .567 \begin{align*}=\end{align*} ______
This problem is going to have several steps to it because we are multiplying decimals that are in the thousandths place.
That is alright though. We can do the same thing that we did with larger decimals. We can multiply the numbers as if they were whole numbers and then insert the decimal point at the end into the final product.
Let’s start by rewriting the problem vertically instead of horizontally.
Example
\begin{align*}.134 \\ \underline{\times \quad \ .567} \\ 938 \\ 8040 \\ \underline{ + \ \ 67000} \\ 75978\end{align*}
Wow! There are a lot of digits in that number-now we need to put the decimal point into the product.
There are three decimal places in the first number .134.
There are three decimal places in the second number .567.
We need to count six decimal places from right to left in the product.
When this happens, we can add a zero in front of the digits to create the sixth place.
.075978
Our final answer is .075978.
Sometimes, we only need to multiply to a specific place. Let’s say that we only wanted to multiply to the ten-thousandths place.
If we were using this example, we would count to the ten-thousandths place in our product and round to the nearest place.
.075978 - the 9 is in the ten-thousandths place
There is a 7 after the nine, so we can round up.
Our final answer is .0760.
Now it is your turn to practice. Multiply each pair of decimals.
- .56 \begin{align*}\times\end{align*} 3.24
- .27 \begin{align*}\times\end{align*} .456
- .18 \begin{align*}\times\end{align*} .320
Stop and check your work.
IV. Solve Real-World Problems Involving Area of Rectangles with Decimal Dimensions
In our last lesson we looked at how to find the area of a rectangle composed of two rectangles using the Distributive Property. This section looks at how to find the area of a rectangle when there are decimal dimensions.
Let’s look at an example.
Example
Jesus wants to put new carpeting down in his bedroom. He measured out the length of the room and found that it was \begin{align*}12 \frac{1}{2}\end{align*} feet long. The width of the room is \begin{align*}9 \frac{1}{2}\end{align*} feet long. Given these dimensions, how many square feet of carpet will Jesus need?
This is a problem that almost everyone will need to solve at one time or another. Whether you are a student redecorating, a college student fixing up a dorm room or an adult remodeling or redesigning a home.
To start with, let’s draw a picture of Jesus’ room.
We use the formula for finding the area of a rectangle when solving this problem.
\begin{align*}A = lw\ (\text{length} \times \text{width})\end{align*}
Next, we can substitute our given dimensions into this formula.
\begin{align*}A = (12.5)(9.5)\end{align*}
We multiply as if these measurements were whole numbers and then add in the decimal point.
\begin{align*}12.5 \\ \underline{\times \qquad 9.5} \\ 625 \\ \underline{+ \quad 11250} \\ 11875\end{align*}
Our final step is to insert the decimal point two decimal places.
Our answer is 118.75 square feet.
Now it’s time for a little practice. Find the area of each rectangle.
1.
2.
3.
Stop and check your work for accuracy. Did you remember to label the measurements correctly?
Real Life Example Completed
The Triceratops Skeleton
Now that you have learned all about multiplying decimals, let’s help Kara figure out the height of the triceratops.
Here is the problem once again.
When the students in Mrs. Andersen’s class arrive at the Science Museum, Kara is very excited to learn that there is a dinosaur exhibit. In fact, it is a famous dinosaur exhibit. A set of dinosaur bones from a triceratops has been reconstructed and is on display.
Kara can’t wait to get to see it. She has a feeling that this is going to be her favorite part of the museum. Several other students are equally excited, so Mrs. Andersen and the chaperones decide to go to the exhibit first and the split up into groups.
When Kara walks in she is delighted. There before her eyes is a huge skeleton of a triceratops. After visiting the exhibit for a while, the students begin to move on. Mrs. Andersen sees Kara hesitate before leaving the exhibit. She walks over to her.
“Imagine, that dinosaur is about 4 and a half times as long as you are!” Mrs. Andersen smiles.
The students exit the exhibit hall, but Kara pauses at the door. She has to think about this. In all of her excitement she forgot to find the information that actually says how tall the triceratops actually is.
Mrs. Andersen’s words stay with her, “the dinosaur is \begin{align*}4 \frac{1}{2}\end{align*} times as long as you are.”
Kara knows that she is \begin{align*}5 \frac{1}{4}\end{align*} feet tall. If the dinosaur is \begin{align*}4 \frac{1}{2}\end{align*} times as long as she is, how long is the dinosaur?
While Mrs. Andersen and the chaperones start to split up the students, Kara begins working some quick math on the back of her museum map.
She writes down the following figures.
5.25 \begin{align*}\times\end{align*} 4.5 \begin{align*}=\end{align*} ______
First, let’s go back and underline all of the important information.
Now let’s work on figuring out the height of the triceratops.
First, let’s estimate the product.
5.25 rounds down to 5.
4.5 rounds up to 5
5 \begin{align*}\times\end{align*} 5 is 25 feet tall.
The triceratops is approximately 25 feet long.
Now let’s figure out its actual height.
\begin{align*}5.25 \\ \underline{\times \quad \ \ 4.5} \\ 2625 \\ \underline{+ \ \ 21000} \\ 23625 \end{align*}
Next, we add in the decimal point.
The triceratops is 23.6 feet long. He is a little longer than 23 and one-half feet.
Wow! That is one big dinosaur!!
Vocabulary
Here are the vocabulary words that can be found in this lesson.
- Hundreds grid
- a grid of one hundred boxes used to show hundredths when working with decimals.
- Product
- the answer in a multiplication problem.
- Vertically
- written up and down in columns
- Horizontally
- written across
- Area
- the surface or space inside a perimeter
Technology Integration
James Sousa Multiplying Decimals
James Sousa Example of Multiplying Decimals
James Sousa Another Example of Multiplying Decimals
Other Videos:
http://www.mathplayground.com/howto_multiplydecimals.html – A good basic video on multiplying decimals
Time to Practice
Directions: Multiply the following decimals.
1. 4.3 \begin{align*}\times\end{align*} .12 \begin{align*}=\end{align*} ______
2. 2.3 \begin{align*}\times\end{align*} 3.4 \begin{align*}=\end{align*} ______
3. .34 \begin{align*}\times\end{align*} .56 \begin{align*}=\end{align*} ______
4. 2.7 \begin{align*}\times\end{align*} 3.2 \begin{align*}=\end{align*} ______
5. 6.5 \begin{align*}\times\end{align*} 2.7 \begin{align*}=\end{align*} ______
6. .23 \begin{align*}\times\end{align*} .56 \begin{align*}=\end{align*} ______
7. 1.23 \begin{align*}\times\end{align*} .4 \begin{align*}=\end{align*} ______
8. .5 \begin{align*}\times\end{align*} .76 \begin{align*}=\end{align*} ______
9. .23 \begin{align*}\times\end{align*} .8 \begin{align*}=\end{align*} ______
10. 3.45 \begin{align*}\times\end{align*} 1.23 \begin{align*}=\end{align*} ______
11. 1.45 \begin{align*}\times\end{align*} .23 \begin{align*}=\end{align*} ______
12. .89 \begin{align*}\times\end{align*} .9 \begin{align*}=\end{align*} ______
13. .245 \begin{align*}\times\end{align*} .8 \begin{align*}=\end{align*} ______
14. 34.5 \begin{align*}\times\end{align*} .7 \begin{align*}=\end{align*} ______
15. 18.7 \begin{align*}\times\end{align*} .9 \begin{align*}=\end{align*} ______
16. 22.3 \begin{align*}\times\end{align*} .76 \begin{align*}=\end{align*} ______
17. 21.7 \begin{align*}\times\end{align*} .4 \begin{align*}=\end{align*} ______
18. 14.5 \begin{align*}\times\end{align*} .68 \begin{align*}=\end{align*} ______
19. 20.1 \begin{align*}\times\end{align*} .3 \begin{align*}=\end{align*} ______
20. 34.23 \begin{align*}\times\end{align*} .18 \begin{align*}=\end{align*} ______
21. .189 \begin{align*}\times\end{align*} .9 \begin{align*}=\end{align*} ______
22. .341 \begin{align*}\times\end{align*} .123 \begin{align*}=\end{align*} ______
23. .451 \begin{align*}\times\end{align*} .12 \begin{align*}=\end{align*} ______
24. .768 \begin{align*}\times\end{align*} .123 \begin{align*}=\end{align*} ______
25. .76 \begin{align*}\times\end{align*} .899 \begin{align*}=\end{align*} ______
Directions: Find the area of the following rectangles. You may round to the nearest hundredth.
26.
27.
28.
29.
30.
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