4.1: Multiplying Decimals and Whole Numbers
Introduction
The Science Museum Field Trip
Mrs. Andersen is planning a field trip to the Science Museum for her sixth grade class. She wants to spend the entire day at the museum and plans to take all twenty-two students with her.
She looks up some information on the internet and finds that a regular price ticket is $12.95 and a student ticket is $10.95. However, when Mrs. Andersen checks out the group rates, she finds that the students can go for $8.95 per ticket at the group student rate.
Because she is a teacher, Mrs. Andersen gets to go for free.
One chaperone receives free admission also. Mrs. Andersen has a total of three chaperones attending the field trip. The other two chaperones will need to pay the regular ticket price. The class has a budget to pay for the chaperones.
Mrs. Andersen assigns Kyle the job of being Field Trip Manager. She hands him her figures and asks him to make up the permission slip. Kyle is glad to do it.
When collection day comes, Kyle collects all of the money for the trip.
Kyle has an idea how much he should collect, what should his estimate be?
Given the student price, how much money does Kyle need to collect if all 22 students attend the field trip?
What is the total cost for all of the students and for the two chaperones?
While Kyle is adding up the money, you have the opportunity to figure out the answers to these two questions.
You will need to use information about multiplying decimals and whole numbers.
Pay close attention during this lesson, see if your answers match Kyle’s by the end of the lesson.
What You Will Learn
In this lesson you will learn how to complete the following tasks:
- Multiply decimals by whole numbers
- Use and compare methods of estimation to check for reasonableness in multiplication of decimals by whole numbers
- Identify and apply the commutative and associative properties of multiplication in decimal operations, using numerical and variable expressions.
- Solve real-world problems involving decimal multiplication
Teaching Time
I. Multiplying Decimals by Whole Numbers
In this lesson you will be learning about how to multiply decimals and whole numbers together. Let’s think about what it means to multiply.
Multiplication is a short-cut for repeated addition. We think about multiplication and we think about groups of numbers. Let’s look at an example.
Example
4 \begin{align*}\times\end{align*} 3 \begin{align*}=\end{align*} 12
With this example, we are saying that we have four groups of three that we are counting or we have three groups of four. It doesn’t matter which way we say it, because we still end up with twelve.
When we multiply decimals and whole numbers, we need to think of it as groups too.
Example
2(.25) = _____
With this example, we are multiplying two times twenty-five hundredths. Remember that when we see a number outside of the parentheses that the operation is multiplication.
We can think of this as two groups of twenty-five hundredths. Let’s look at what a picture of this would look like.
Our answer is .50.
This is one way to multiply decimals and whole numbers; however we can’t always use a drawing. It just isn’t practical.
How can we multiply decimals and whole numbers without using a drawing?
We can multiply a decimal and a whole number just like we would two whole numbers.
First, we ignore the decimal point and just multiply.
Then, we put the decimal point in the product by counting the correct number of places.
Let’s look at an example.
Example
4(1.25) = _____
Let’s start by multiplying just like we would if this was two whole numbers. We take the four and multiply it by each digit in the top number.
\begin{align*}125 \\ \underline{\times \ \quad 4}\\ 500\end{align*}
But wait! Our work isn’t finished yet. We need to add the decimal point into the product.
There were two decimal places in our original problem. There should be two decimal places in our product.
\begin{align*}& 5.00 \\ & \ \nwarrow \\ & \quad \text{We count in two places from right to left into our product}.\end{align*}
This is our final answer.
Here are a few for you to try. Multiply them just as you would whole numbers and then put in the decimal point.
- 3(4.52)
- 5(2.34)
- 7(3.56)
Take a few minutes to check your work with a neighbor. Did you put the decimal point in the correct place?
II. Use and Compare Methods of Estimation to Check for Reasonableness in Multiplying Decimals by Whole Numbers
We have learned how to multiply a decimal with a whole number. That is the perfect thing to do if you are looking for an exact answer.
When do we estimate a product?
Remember back to when we were first working with estimation. We can use estimation whenever we don’t need to find an exact answer. As long as our answer makes sense, we can estimate.
We can use rounding to estimate.
How can we estimate a product using rounding?
When we multiply a whole number with a decimal, we can round the decimal that we are multiplying to find a reasonable estimate.
Let’s look at an example.
Example
Estimate 5(1.7) = _____
In this example we were told that we could estimate, so we don’t need to worry about finding an exact answer.
If we use rounding, we can round the decimal to the nearest whole number.
1.7 is closest to 2.
We round 1.7 up to 2.
Now we can rewrite the problem and multiply.
Example
5(2) = 10
A reasonable estimate for 5(1.7) is 10.
Here is another example.
Example
Estimate 7(4.3) = _____
Here we can estimate by rounding the decimal.
4.3 rounds down to 4
7 \begin{align*}\times\end{align*} 4 \begin{align*}=\end{align*} 28
A reasonable estimate for 7(4.3) = 28
Here are a few for you to try. Estimate the following products.
- 4(3.2) = _____
- 6(2.8) = _____
- 7(5.3) = _____
Stop and check your answers with a peer. Are your estimates reasonable?
III. Identify and Apply the Commutative and Associative Properties of Multiplication in Decimal Operations using Numerical and Variable Expressions
We have already learned about using the properties of multiplication in numerical and variable expressions. Now we are going to apply these properties to our work with multiplying decimals and whole numbers.
What is a property?
A property is a rule that makes a statement about the way that numbers interact with each other during certain operations. The key thing to remember about a property is that the statement is true for any numbers.
The Commutative Property of Multiplication
The Commutative Property of Multiplication states that it does not matter which order you multiply numbers in, that you will get the same product.
\begin{align*}a(b) = b(a)\end{align*}
What does this have to do with our work with decimals and whole numbers?
When we apply the Commutative Property of Multiplication to our work with decimals and whole numbers, we can be sure that the product will be the same regardless of whether we multiply the decimal first or the whole number first.
Let’s look at an example.
Example
4.5(7) is the same as 7(4.5)
This means that we can multiply them in whichever order we choose. Our product will remain the same.
\begin{align*}45 \\ \underline{\times \quad 7} \\ 315\end{align*}
Add in the decimal point.
Our answer is 31.5.
We can also apply the Commutative Property of Multiplication when we have a problem with a variable in it.
Remember that a variable is a letter used to represent an unknown.
Let’s look at an example.
Example
\begin{align*}5.6a = a5.6\end{align*}
Here we haven’t been given a value for a, but that doesn’t matter. The important thing is for you to see that it doesn’t matter which order we multiply, the product will be the same.
If we were given 3 as the value for a, what would our product be?
Example
\begin{align*}5.6(3)\end{align*}
\begin{align*}56 \\ \underline{\times \ \ 3} \\ 168\end{align*}
Add in the decimal point.
Our answer is 16.8.
The Associative Property of Multiplication
We can also apply the Associative Property of Multiplication to our work with decimals and whole numbers.
The Associative Property of Multiplication states that it doesn’t matter how you group numbers, that the product will be the same.
Remember that grouping refers to the use of parentheses or brackets.
Let’s look at an example of the Associative Property of Multiplication with numbers.
Example
6(3.4 \begin{align*}\times\end{align*} 2) \begin{align*}=\end{align*} (6 \begin{align*}\times\end{align*} 3.4)2
We can change the grouping of the numbers and the product will remain the same.
This is also true when we have variable expressions.
Example
\begin{align*}5(6a) = (5 \times 6)a\end{align*}
Once again, we can change the grouping of the numbers and variables, but the product will remain the same.
Look at these examples and determine which property is being illustrated.
- 4.5(5a) \begin{align*}=\end{align*} (4.5 \begin{align*}\times\end{align*} 5)a
- 6.7(4) = 4(6.7)
- 5.4a = a5.4
Take a few minutes to check your work with a peer.
Real life Example Completed
The Science Museum Field Trip
Now that you have learned all about estimating and multiplying whole numbers and decimals, let’s look at helping Kyle with the field trip.
Here is the problem once again.
Mrs. Andersen is planning a field trip to the Science Museum for her sixth grade class. She wants to spend the entire day at the museum and plans to take all twenty-two students with her.
She looks up some information on the internet and finds that a regular price ticket is $12.95 and a student ticket is $10.95. However, when Mrs. Andersen checks out the group rates, she finds that the students can go for $8.95 per ticket at the group student rate.
Because she is a teacher, Mrs. Andersen gets to go for free.
One chaperone receives free admission also. Mrs. Andersen has a total of three chaperones attending the field trip. The other two chaperones will need to pay the regular ticket price. The class has a budget to pay for the chaperones.
Mrs. Andersen assigns Kyle the job of being Field Trip Manager. She hands him her figures and asks him to make up the permission slip. Kyle is glad to do it.
When collection day comes, Kyle collects all of the money for the trip.
Kyle has an idea how much he should collect, what should his estimate be?
Given the student price, how much money does Kyle need to collect if all 22 students attend the field trip?
What is the total cost for all of the students and for the two chaperones?
First, let’s go back and underline all of the important information.
Now, let’s think about the estimate. About how much money should Kyle collect?
The first step in working this out is to write an equation.
22 students at $8.95 per ticket = 22(8.95)
Kyle wants an estimate, so we can round 8.95 to 9
Now let’s multiply 22(9) = $198.00
Now that Kyle has an estimate, he can actually work on collecting the money and counting it. Once he has collected and counted all the money, we will be able to see if his original estimate was reasonable or not.
One week before the trip, Kyle collects $8.95 from 22 students.
He multiplies his results, 22(8.95) = $196.90
Kyle can see that his original estimate was reasonable. He is excited-the estimation worked!!
Next, Kyle figures out the cost of the chaperones. There are two chaperones who each pay the regular price which is $12.95.
2(12.95) = 25.90
Finally, Kyle adds up the total.
196.90 + 25.90 = $222.80
He gives his arithmetic and money to Mrs. Andersen. She is very pleased.
The students are off to the Science Museum!!!
Vocabulary
Here are the vocabulary words that can be found in this lesson.
- Multiplication
- a shortcut for addition, means working with groups of numbers
- Product
- the answer from a multiplication problem
- Estimate
- an approximate answer-often found through rounding
- Properties
- rules that are true for all numbers
- The Commutative Property of Multiplication
- it doesn’t matter which order you multiply numbers, the product will be the same.
- The Associative Property of Multiplication
- it doesn’t matter how you group numbers in a multiplication problem, the product will be the same.
Technology Integration
Khan Academy Multiplying Decimals 2
This video presents multiplying decimals by whole numbers. http://www.youtube.com/watch?v=EZ4KI0pv4Fk
Time to Practice
Directions: Estimate the following products.
1. 4(3.2) = _____
2. 5(1.8) = _____
3. 6(2.3) = _____
4. 9(1.67) = _____
5. 8(4.5) = _____
6. 9(6.7) = _____
7. 4(8.1) = _____
8. 8(3.2) = _____
9. 9(9.7) = _____
10. 7(1.1) = _____
11. 8(3.5) = _____
12. 5(8.4) = _____
Directions: Multiply to find a product.
13. 5(1.24) = _____
14. 6(7.81) = _____
15. 7(9.3) = _____
16. 8(1.45) = _____
17. 9(12.34) = _____
18. 2(3.56) = _____
19. 6(7.12) = _____
20. 3(4.2) = _____
21. 5(2.4) = _____
22. 6(3.521) = _____
23. 2(3.222) = _____
24. 3(4.223) = _____
25. 4(12.34) = _____
26. 5(12.45) = _____
27. 3(143.12) = _____
28. 4(13.672) = _____
29. 2(19.901) = _____
30. 3(67.321) = _____
Directions: Identify the property illustrated in each example.
31. 4.6a = a4.6
32. (4a)(b) = 4(ab)
33. (5.5a)(c) = 5.5(ac)