5.1: Multiplying by 10

In which equation is the value of $x$ the largest?
$x \boldcdot 10 = 810$
$x \boldcdot 10 = 81$
$x \boldcdot 10 = 8.1$
$x \boldcdot 10 = 0.81$

How many times the size of 0.81 is 810?
Let’s look at products that are decimals.
In which equation is the value of $x$ the largest?
$x \boldcdot 10 = 810$
$x \boldcdot 10 = 81$
$x \boldcdot 10 = 8.1$
$x \boldcdot 10 = 0.81$
How many times the size of 0.81 is 810?
Work with a partner to answer the following questions. One person should answer the questions labeled “Partner A,” and the other should answer those labeled “Partner B.” Then compare the results.
Find each product or quotient. Be prepared to explain your reasoning.
Partner A
Partner B
Find each product. Show your reasoning.
$36 \boldcdot (0.1)$
$(24.5) \boldcdot (0.1)$
$(1.8) \boldcdot (0.1)$
$54 \boldcdot (0.01)$
$(9.2)\boldcdot (0.01)$
Select all expressions that are equivalent to $(0.6) \boldcdot (0.5)$. Be prepared to explain your reasoning.
Find the value of each product by writing and reasoning with an equivalent expression with fractions.
Ancient Romans used the letter I for 1, V for 5, X for 10, L for 50, C for 100, D for 500, and M for 1,000.
We can use fractions like $\frac{1}{10}$ and $\frac{1}{100}$ to reason about the location of the decimal point in a product of two decimals.
Let’s take $24 \boldcdot (0.1)$ as an example. There are several ways to find the product:
Similarly, we can think of $(0.7) \boldcdot (0.09)$ as 7 tenths times 9 hundredths, and write:
$$\left(7 \boldcdot \frac {1}{10}\right) \boldcdot \left(9 \boldcdot \frac {1}{100}\right)$$
We can rearrange whole numbers and fractions:
$$(7 \boldcdot 9) \boldcdot \left( \frac {1}{10} \boldcdot \frac {1}{100}\right) = 63 \boldcdot \frac {1}{1,\!000} = \frac {63}{1,\!000}$$
This tells us that $(0.7) \boldcdot (0.09) = 0.063$.
Here is another example: To find $(1.5) \boldcdot (0.43)$, we can think of 1.5 as 15 tenths and 0.43 as 43 hundredths. We can write the tenths and hundredths as fractions and rearrange the factors. $$\left(15 \boldcdot \frac{1}{10}\right) \boldcdot \left(43 \boldcdot \frac{1}{100}\right) = 15 \boldcdot 43 \boldcdot \frac{1}{1,\!000}$$
Multiplying 15 and 43 gives us 645, and multiplying $\frac{1}{10}$ and $ \frac{1}{100}$ gives us $\frac{1}{1,000}$. So $(1.5) \boldcdot (0.43)$ is $645 \boldcdot \frac{1}{1,000}$, which is 0.645.