In this lesson, we'll expand our understanding of equivalent place values.
In a previous lesson, we saw that
We can picture this fact using a diagram:
How can we use this to express " tens" in terms of hundreds?
The first thing to realize is that
Now, since " tens" is the same as " hundred", we have
So, we conclude that
This equivalence can also be expressed using a diagram:
Using similar arguments, we're able to create the following pattern:
tens is equivalent to hundred
tens is equivalent to hundreds
tens is equivalent to hundreds
tens is equivalent to hundreds
tens is equivalent to hundreds
tens is equivalent to hundreds
By combining this pattern with our understanding of expanded form, we can find more equivalent place value representations.
Let's see an example.
What is " tens" expressed in terms of hundreds and tens?
Since we can write tens as
Now, since tens is the same as hundreds, the statement above is equal to
Therefore, " tens" is equivalent to " hundreds and tens".
"$51$ tens" is equivalent to
|
a
|
$5,100$ ones |
|
b
|
$5$ hundreds and $1$ ten |
|
c
|
$5$ tens and $1$ one |
|
d
|
$5$ hundreds and $1$ one |
|
e
|
$5$ thousands and $1$ ten |
"$90$ tens" is equivalent to
|
a
|
$9$ hundreds |
|
b
|
$9$ hundreds and $9$ tens |
|
c
|
$9$ thousands |
|
d
|
$9$ hundreds and $9$ ones |
|
e
|
$9$ ones |
Using similar arguments to what we saw previously, we're able to establish the following pattern for multiple hundreds.
hundreds is equivalent to thousand
hundreds is equivalent to thousands
hundreds is equivalent to thousands
hundreds is equivalent to thousands
hundreds is equivalent to thousands
hundreds is equivalent to thousands
Similar patterns exist for multiple thousands, ten-thousands, hundred-thousands, etc.
Express " hundreds" in terms of thousands and hundreds.
Since we can write hundreds as
Now, since hundreds is the same as thousands, the statement above is equal to
Therefore, " hundreds" is equivalent to " thousands and hundreds".
"$59$ hundreds" is equivalent to
|
a
|
$5$ thousands and $9$ hundreds |
|
b
|
$5$ ten-thousands and $9$ thousands |
|
c
|
$5$ thousands and $9$ ones |
|
d
|
$5$ thousands and $9$ tens |
|
e
|
$5$ ten thousands and $9$ tens |
"$40$ hundreds" is equivalent to
|
a
|
$4$ thousands |
|
b
|
$4$ thousands and $4$ hundreds |
|
c
|
$4$ hundreds |
|
d
|
$4$ ten thousands |
|
e
|
$4$ hundreds and $4$ tens |
When there are more than two digits in one place value, it's usually easier to use a place value chart to find an equivalent representation.
For example, let's find an equivalent place value representation for " hundreds".
We start with its place value chart:
| ten-thousands | thousands | hundreds | tens | ones |
There are three digits in the "hundreds" column. Therefore,
we move the first digit two places to the left (to the "ten-thousands" column), and
we move the second digit one place to the left (to the "thousands" column).
This gives the following place value chart:
| ten-thousands | thousands | hundreds | tens | ones |
Therefore, " hundreds" is equivalent to " ten-thousands and hundred."
Express " tens" in terms of thousands, hundreds, and tens.
Let's write " tens" in a place value chart:
| thousands | hundreds | tens | ones |
There are three digits in the "tens" column. Therefore,
we move the first digit two places to the left (to the "thousands" column), and
we move the second digit one place to the left (to the "hundreds" column).
This gives the following place value chart:
| thousands | hundreds | tens | ones |
Therefore, " tens" is equivalent to " thousands, hundreds, and tens."
"$512$ tens" is equivalent to
|
a
|
$51$ thousands and $2$ tens |
|
b
|
$5$ thousands, $1$ hundred, and $2$ tens |
|
c
|
$5$ thousands, $1$ ten, and $2$ one |
|
d
|
$51$ thousands and $2$ ones |
|
e
|
$5$ hundreds and $12$ tens |
"$406$ hundreds" is equivalent to
|
a
|
$4$ hundred-thousands and $6$ tens |
|
b
|
$4$ ten-thousands and $6$ hundreds |
|
c
|
$4$ hundred-thousands and $6$ hundreds |
|
d
|
$4$ ten-thousands and $6$ tens |
|
e
|
$4$ hundred-thousands and $6$ thousands |