strategies for multiplying decimal numbers

Decimal Times

Here are two everyday situations in which you could multiply decimals .
• How many square feet of carpet do you need to carpet
an 11.5 by 12.5 foot room?
• One soccer ball costs $16.79.You buy four soccer balls.
How can you find the total cost?

In this investigation you will develop strategies for multiplying decimal
numbers. Estimating can help you decide whether a product is reasonable .
Changing the form of a number from a fraction to a decimal , or a decimal
to a fraction, can help you estimate or find an exact answer.

2.1 Relating Fraction and Decimal Multiplication

You know you can write decimals as fractions. You can also write fractions
as decimals. For example, to write 2/5 as a decimal, first rewrite it as the
equivalent fraction 4/10.Then write 4/10 as the decimal 0.4.


The grid on the left below is a tenths grid with one strip shaded. This strip
re presents 1/10 or 0.1. On the right, the strip representing 0.1 is divided into
10 squares . The single square with darker shading is 1/10 of 1/10,or 0.1 × 0.1.

You can extend the horizontal lines to make a hundredths grid.
This shows that 1/10 of 1/10 is one square out of one hundred squares,
which is 1/100 or 0.01 of the whole.

This makes sense because decimals are just fractions written in a
different notation . You know that 0.63 = 63/100 and that 0.0063 = 63/10000.
Fraction multiplication can help you with decimal multiplication.

Getting Ready for Problem 2.1

To find the product of 0.3 × 2.3, you can use equivalent fractions.

• What is the product written as a fraction?
• What is the product written as a decimal?
• How can knowing the product as a fraction help you write the product
in decimal form?

Problem 2.1 Relating Fraction and Decimal Multiplication

A. On the next page are some multiplication situations that happen at the
Apple-A-Day fruit stand. For each situation, do the following:
• Estimate the product.
• Write the decimals in fraction notation with denominators of 10,
100, 1000, etc.
• Find the exact answer and compare it to your estimate.

1. Ajay buys 1.7 pounds of Red Delicious apples on sale for $0.50 a
pound. What is his bill?
2. Kelly buys 0.4 of a pound of Granny Smith apples on sale for
$0.55 a pound. What is her bill?
3. Chayton buys Northern Spy apples for his mother’s pie. He buys
3.2 pounds at $1.10 a pound. What is his bill?

4. The Apple-A-Day roadside fruit stand sells bunches of wildflowers.
The seller wraps each bunch in a sheet of paper with dimensions of
1.5 feet by 1.5 feet. What is the area of the paper?

B. Why does it make sense to use multiplication to solve the situations in
Question A?

C.
1. When one factor in a multiplication problem is greater than 1, is
the product greater or less than the other factor? Explain.
2. When one factor is less than 1, is the product greater or less than
the other factor?

2.2Missing Factors

Suppose you put decimal points into 125 × 5 to make the problem
1.25 × 0.5.
• How is the product of 125 × 5 related to the product of 1.25 × 0.5?
• Does writing the problem as help?

Problem 2.2 Missing Factors

A. Use what you know about fraction multiplication and place value.
1. What number times 6 gives the product 0.36?
2. What number times 0.9 gives the product 2.7?
3. What number times 1.5 gives the product 0.045?
4. What strategies did you use to solve these problems?

B. Use what you know about decimal multiplication and place value.
1. Find two numbers with a product of 1,344.
2. Find two numbers with a product of 134.4.
3. Find two numbers with a product of 0.1344.
4. What strategies did you use to solve these problems?

C. 1. What number times 0.3 gives the product 9?
2. What number times 0.12 gives the product 24?
3. What strategies did you use to solve these problems?

2.3Finding Decimal Products

Sometimes you don’t need to know an actual product. Estimation can help
you find an amount that is reasonable.

Sometimes an estimate can help you decide if a computation is correct.

Sometimes an estimate can help you decide where to place the decimal
point in an actual product.

Getting Ready for Problem 2.3

Look at two different estimation strategies for the problem 2.1 × 1.4.
Jose rounded to whole numbers and fractions:
Rosa only rounded one number: 2 × 1.4 = 2.8
• Are both estimates reasonable? Which estimate is closer to the actual
answer?

Problem 2.3 Finding Decimal Products

A. Estimate each product. Describe how you found your estimate.
1. 0.9 ×3.4
2. 4.92 ×0.5
3. 0.22 × 0.301
4. 23.87 × 6.954

B. Julia says that sometimes she uses
estimation to decide where to place
the decimal in an actual product.
With the problem 0.9 × 1.305, a
reasonable estimate is 1 × 1.3 = 1.3.
Even 1 × 1 is a good estimate. I think
that the actual product is a little more
than 1. When I multiply 9 × 1,305
I get 11,745, so I know the actual
product is 1.1745.

1. How did Julia use her estimate to find the actual product of
1.1745?
2. Use Julia’s estimation strategy to find the product N.
a. 31.2 × 2.1 = N
b. If 6,946 × 28 = 194,488, then 694.6 × 2.8 = N.

2.4 Factor–Product Relationships

Understanding our place value system depends on knowing about
The numbers 10; 100; 1,000; 10,000; and so on are powers of ten .
You can write these numbers as 10, 10 × 10, 10 × 10 × 10, and so on OR as
10, 10², 10³. . . .

Our place value system is centered around the units place (ones). A number
that is one place to the left of the units place has a value ten times larger.
For example, in the number 77, the 7 on the right represents 7 and the 7 on
the left represents 7 × 10, or 70.

As we move to the right we divide by powers of ten.

Problem 2.4 Factor–Product Relationships

Estimate before you find an exact answer to the questions below.
A. Record the products in each set in an organized way. Describe any
patterns that you see.



B. In a decimal multiplication problem, there is a relationship between
the number of decimal places in the factors and the number of decimal
places in the product.
1. Use your answers to Question A to summarize what you think the
decimal place relationship is.
2. Test the relationship on these two problems:
a. 4.5 × 0.9
b. 0.004 × 0.12
3.
a. Write the two problems in part (2) as fractions. Find the
product using fraction multiplication.
b. How does fraction multiplication support the relationship you
tested in part (2)?

C. Describe an algorithm you can use to multiply any two decimal
numbers.
D.
1. Find the following products using the fact that 21 ×11 = 231.

2. Test the algorithm you wrote in Question C on the problems.