Topic A: Multiplicative patterns on the place value chart

##Place value chart

The number $87.654$ can be placed on a place value chart as follows:

##Decimal place value

The first digit after the decimal represents the tenths place. The next digit after the decimal represents the hundredths place. The remaining digits continue to fill in the place values until there are no digits left.

Example: $0.81$

The number $0.\blueD8\greenD1$ is made up of $\blueD8$ tenths and $\greenD1$ hundredth.

We can also write this as:

$0.\blueD8\greenD1=\blueD{0.8}+\greenD{0.01}$

or

$0.\blueD8\greenD1=\blueD{\dfrac8{10}}+\greenD{\dfrac1{100}}$

Or we can use a place value chart:

Want to learn more about decimal place value? Check out this video.

Want to review whole number place value? Check out this article.

##Practice

[[☃ graded-group-set 2]]

Want to try more problems like this? Check out these exercises:
Decimal place value names
Value of a decimal digit

##Summary of powers of $10$

##How do we evaluate powers of $10$?

To evaluate a power of $10$, we multiply the base $(10)$ by itself the exponent amount of times.

Example:

$10^5$ has an exponent of $5$, so we multiply $10$ times itself $5$ times:

$10^5=10\times10\times10\times10\times10=100{,}000$

##The pattern

The number of zeros in the solution always matches the exponent.

Example:

$10^\greenD{3}=10\times10\times10=1\greenD{000}$, which has $\greenD3$ zeros

Want to learn more about powers of ten? Check out this video.

##Practice

[[☃ graded-group-set 1]]

Want to try more problems like this? Check out this exercise.

##Part 1: Multiplying decimals by 10

Key idea: Multiplying by 10 moves every digit one place to the left.

[[☃ graded-group 1]]

[[☃ explanation 1]]

##Let's try a few more.

[[☃ graded-group 1]]

[[☃ graded-group 2]]

[[☃ graded-group 3]]

[[☃ explanation 1]]

##Part 2: Multiplying decimals by 100

Key idea: Multiplying by 100 moves every digit two places to the left.

[[☃ graded-group 2]]

[[☃ explanation 1]]

##Let's try a few more.

[[☃ graded-group 1]]

[[☃ graded-group 2]]

[[☃ graded-group 3]]

[[☃ explanation 1]]

##Part 3: Multiplying decimals by 1,000

Key idea: Multiplying by 1,000 moves every digit three places to the left.

[[☃ graded-group 1]]

[[☃ explanation 1]]

##Let's try a few more.

[[☃ graded-group 1]]

[[☃ graded-group 2]]

[[☃ graded-group 3]]

[[☃ explanation 1]]

##Part 4: Let's look at the pattern.

[[☃ graded-group 1]]

[[☃ explanation 1]]

Part 5: Challenge time!

Use the pattern above to answer the following questions.

[[☃ graded-group 2]]

[[☃ graded-group 3]]

[[☃ explanation 2]]

Part 1: Dividing decimals by 10

Key idea: Dividing by ${10}$ moves every digit one place to the right.

[[☃ graded-group 1]]

[[☃ explanation 1]]

##Let's try a few more dividing by $10$:

[[☃ graded-group 1]]

[[☃ graded-group 3]]

[[☃ graded-group 4]]

[[☃ explanation 1]]

And one multiplying by $10$ (for review):

[[☃ graded-group 2]]

[[☃ explanation 2]]

Part 2: Dividing decimals by 100

Key idea: Dividing by ${100}$ moves every digit two places to the right.

[[☃ graded-group 1]]

[[☃ explanation 1]]

##Let's try a few more dividing by $100$:

[[☃ graded-group 1]]

[[☃ graded-group 3]]

[[☃ graded-group 4]]

[[☃ explanation 1]]

And one multiplying by $100$ (for review):

[[☃ graded-group 2]]

[[☃ explanation 2]]

Part 3: Dividing decimals by 1000

Key idea: Dividing by ${1000}$ moves every digit three places to the right.

[[☃ graded-group 6]]

[[☃ explanation 1]]

##Let's try a few more dividing by $1000$:

[[☃ graded-group 6]]

[[☃ graded-group 8]]

[[☃ graded-group 9]]

[[☃ explanation 1]]

And one multiplying by $1000$ (for review):

[[☃ graded-group 7]]

[[☃ explanation 2]]

Part 4: Let's look at the pattern.

[[☃ graded-group 3]]

[[☃ explanation 3]]

Part 5: Challenge time!

Use the pattern above to answer the following questions.

[[☃ graded-group 1]]

[[☃ graded-group 3]]

[[☃ explanation 1]]

And a multiplication challenge:

[[☃ graded-group 2]]

[[☃ explanation 2]]