Tenths of (and So On)

Task

Since $0.1 = \frac{1}{10}$, when we multiply by 0.1, we are multiplying by one-tenth. 

1. Multiply:

• $0.1 \times 100$
• $0.1 \times 10$
• $0.1 \times 1$
• $0.1 \times 0.1$
• $0.1 \times 0.01$
• $0.1 \times 0.001$

2. Describe the patterns you see in the products above.

 

Similarly, since $0.01 = \frac{1}{100}$, when we multiply by 0.01, we are multiplying by one-hundredth. 

3. Multiply:

• $0.01 \times 100$
• $0.01 \times 10$
• $0.01 \times 1$
• $0.01 \times 0.1$
• $0.01 \times 0.01$
• $0.01 \times 0.001$

4. Describe the patterns you see in the products above.

Based only on the patterns above, what do you expect $0.0001 \times 0.00001$ to be? Explain why that must be true by thinking of these decimals as fractions.

IM Commentary

The purpose of this task is to extend students' understanding of products of decimals by first focusing on products of unit fractions of the form $\frac{1}{10^n}$, where $n$ is a positive integer. Students should have strategies from grade 5 for finding all of the products to the hundredths (5.NBT.B.7) that should easily extend to finding all of the products in this task. These strategies include writing the decimals as fractions and multiplying the numerators and denominators or drawing diagrams. Students might also reason that they know that a digit that is in one place is one-tenth as big as if it were the same digit one place to the left (5.NBT.A.1), and find the first set of products by noting that each product is one-tenth of the right-hand factor; similar reasoning for the second list can be used by noting that a hundredth is a tenth of a tenth. Students who reason along these lines are engaging in MP7, look for and make use of structure.

Patterns students might see include noting that the first set of products are all $\frac{1}{10}$ as big as the previous product in the list and the second set of products are all $\frac{1}{100}$ as big as the previous product in that list, or they might notice that the number of decimal places goes up by one in the first set and up by two in the second set as they go down the list. Ideally, students will make the connection that when we multiply by $\frac{1}{10^n}$, the decimal point moves over $n$ units to the left, or equivalently, the 1 moves $n$ places to the right. Students who see and articulate these patterns are engaging in MP8, look for and express regularity in repeated reasoning.

A whole-class discussion that connects the reasoning students did to find the products with the pattern they see will help them make sense of the rule for placing a decimal point in a product of decimals. One wrinkle that will likely come up (and may need to be explained and demonstrated on a few different occasions): In $0.01 \times 0.001$, for example, the rule based on moving a decimal point would say start with $0.001$ and move the decimal point two places to the left. However, there is only one zero to the left of the decimal point. Students need to know they can move the decimal point as many "spaces" as they want, and fill in any empty spaces with zeroes. Or, they could think of $0.001$ as having as many leading zeroes as you want, like $000000.001$ before they start moving the decimal point to the left.

Solution

(Note: the work in the middle column represents productive scratchwork, but is not a necessary part of the solution. Teachers might suggest students write out these products if they are struggling to get started.)

$0.1 \times 100 =$	$ \frac{1}{10} \times 100 =$	10
$0.1 \times 10 =$	$ \frac{1}{10} \times 10 =$	1
$0.1 \times 1 =$	$ \frac{1}{10} \times 1 =$	0.1
$0.1 \times 0.1 =$	$\frac{1}{10} \times \frac{1}{10} = \frac{1}{100} =$	0.01
$0.1 \times 0.01 =$	$\frac{1}{10} \times \frac{1}{100} = \frac{1}{1000} =$	0.001
$0.1 \times 0.001 =$	$\frac{1}{10} \times \frac{1}{1000} = \frac{1}{10000} =$	0.0001

One of many patterns is that the 1 digit in the product is one place to the left of its place in the right-hand factor. Multiplying by 0.1 appears to "move the decimal point one place to the left."

$0.01 \times 100 =$	$ \frac{1}{100} \times 100 =$	1
$0.01 \times 10 =$	$ \frac{1}{100} \times 10 =$	0.1
$0.01 \times 1 =$	$ \frac{1}{100} \times 1 =$	0.01
$0.01 \times 0.1 =$	$\frac{1}{100} \times \frac{1}{10} = \frac{1}{1000} =$	0.001
$0.01 \times 0.01 =$	$\frac{1}{100} \times \frac{1}{100} = \frac{1}{10000} =$	0.0001
$0.01 \times 0.001 =$	$\frac{1}{100} \times \frac{1}{1000} = \frac{1}{100000} =$	0.00001

One of many patterns is that the 1 digit in the product is two places to the right of its place in the right-hand factor. Multiplying by 0.01 appears to "move the decimal point two places to the left."

Based on these patterns, perhaps $0.0001\times0.00001$ is 0.000000001. Let's check:

$$0.0001\times0.00001 = \frac{1}{10000}\times\frac{1}{100000} = \frac{1}{1000000000}$$

Which really is 0.000000001.