1. Place Value: Expanding Whole Numbers 3 Ways
For Dixon Elementary School’s
5th-Grade Math Classes

2. Expanded Forms
When expanding numbers, there are 3 different methods we can use:
expanding by place value,
expanding by base-10 products,
and
expanding by base-10 exponents.
You might see 1, 2, or all 3 methods on standardized tests.

3. Expanded Form by Place Value
Method #1 of expanding shows numbers expanded into the sum (+) of each digit’s place value.
This method just involves decomposing numbers.
The number 652 would expand like this:
The value of the 6 = 600, the value of the 5 = 50, and the value of the 2 = 2.

4. Expanded Form by Place Value
Here’s another example: 4,365
There is a 4 in the one thousands place, so the value of the 4 = 4,000 (4 x 1,000).
There is a 3 in the hundreds place, so the value of the 3 = 300 (3 x 100).
There is a 6 in the tens place, so the value of the 6 = 60 (6 x 10).
There is a 5 in the ones place, so the value of the 5 = 5 (5 x 1). 4,

5. Expanded Form by Place Value
When expanding numbers with 0s in a place, we can skip that place, and move on to the next:
60,503 =
60,

6. Expanded Form by Place Value: Your Turn!
Expand each number by its place value: 79
893
1,265
20,482
304,007
ANSWERS:
70 + 9 1,
20,
300, ,

7. Expanded Notation with Base-10 Products
Method #2 of expanding involves using base-10 products with expanded notation.
Base-10 is the number system we use in math.
The system uses 10 numerical symbols to form all numbers:
0, 1, 2, 3, 4, 5, 6, 7, 8, & 9
When expanding with base-10 products, we call it expanded notation.

8. Expanded Notation with Base-10 Products
In expanded notation, we write the number as the sum (+) of the products of each digit multiplied by the value of its place.
The value of the ONES place = 1.
The value of the TENS place = 10.
The value of the HUNDREDS place = 100.
What is the value of the THOUSANDS place?

9. Expanded Notation with Base-10 Products
Let’s look at the number, 359.
The 3 is in the hundreds place, so we multiply the 3 by 100 (3 x 100).
The 5 is in the tens place, so we multiply the 5 by 10 (5 x 10).
The 9 is in the ones place, so we multiply the 9 by 1 (9 x 1).
So the expanded notation is:
(3 x 100) + (5 x 10) + (9 x 1)

10. Expanded Notation with Base-10 Products
Here are some more examples:
6,219 =
(6 x 1,000) + (2 x 100) +
(1 x 10) + (9 x 1)
54,678 =
(5 x 10,000) + (4 x 1,000) +
(6 x 100) + (7 x 10) + (8 x 1)

11. Expanded Notation with Base-10 Products
Let’s do this one together:
3,462 =
The 3 is in the place, so we multiply the 3 by
The 4 is in the place, so we multiply the 4 by
The 6 is in the place, so we multiply the 6 by
The 2 is in the place, so we multiply the 2 by
thousands ? ?
1,000
hundreds ? ?
100 ?
tens 10 ? ?
ones 1 ?

12. Expanded Notation with Base-10 Products
So…
3,462 =
(3 x ) + (4 x ) +
(6 x ) + (2 x ) ?
1,000
100 ? 10 ? 1 ?

13. Expanded Notation with Base-10 Products: Your Turn!
Use expanded notation to expand each number.
1) 48
2) 387
3) 6,912
4) 52,376

14. Expanded Notation with Base-10 Products: Your Turn!
ANSWERS:
1) 48 = (4 x 10) + (8 x 1)
2) 387 = (3 x 100) + (8 x 10) + (7 x 1)
3) 6,912 = (6 x 1,000) + (9 x 100) +
(1 x 10) + (2 x 1)
4) 52,376 = (5 x 10,000) + (2 x 1,000) +
(3 x 100) + (7 x 10) + (6 x 1)

15. Expanded Notation with Base-10 Exponents
Method #3 of expanding is expanded notation with exponents.
An exponent shows the number of times a number is multiplied by itself.
An exponent is attached to its base number like this:
103
exponent
base

16. Expanded Notation with Base-10 Exponents
We say 103 like this:
10 to the power of 3 OR
10 to the 3rd power.
What 103 means is:
10 x 10 x 10
(10 multiplied by itself 3 times)

17. Expanded Notation with Base-10 Exponents
Here are some more examples:
1) 102: Say, “10 to the power of 2” or
“10 to the 2nd power.”
It means, 10 x 10
2) 104: Say, “10 to the power of 4” or
“10 to the 4th power.”
It means, 10 x 10 x 10 x 10

18. Expanded Notation with Base-10 Exponents: Your Turn!
How do we SAY these bases & exponents in 2 different ways?
1) 106
2) 101
3) 104
4) 105
5) 107
10 to the power of 6 & 10 to the 6th power
10 to the power of 1 & 10 to the 1st power
10 to the power of 4 & 10 to the 4th power
Complete this practice orally.
10 to the power of 5 & 10 to the 5th power
10 to the power of 7 & 10 to the 7th power

19. Expanded Notation with Base-10 Exponents: Your Turn!
What does each MEAN?
1) 106
2) 101
3) 104
4) 105
5) 107

20. Expanded Notation with Base-10 Exponents: Your Turn!
ANSWERS:
106 = 10 x 10 x 10 x 10 x 10 x 10
2) 101 = 10
3) 104 = 10 x 10 x 10 x 10
4) 105 = 10 x 10 x 10 x 10 x 10
5) 107 = 10 x 10 x 10 x 10 x 10 x 10 x 10

21. Expanded Notation with Base-10 Exponents
To calculate the value of a base and its exponent, just complete the multiplication:
102 = 10 x 10,
and 10 x 10 = 100,
so 102 = 100;
103 = 10 x 10 x 10,
and 10 x 10 x 10 = 1,000,
so 103 = 1,000.

22. Expanded Notation with Base-10 Exponents
Here are some more examples:
1) 104 = 10 x 10 x 10 x 10 = 10,000
2) 105 = 10 x 10 x 10 x 10 x 10 =
100,000
Do you notice a relationship between the exponent and its number’s value?
104 = 10,000
105 = 100,000

23. Expanded Notation with Base-10 Exponents
The relationship between the exponent and its number’s value makes finding its value easy!
100 = 1
101 = 10
102 = 100
103 = 1,000
104 = 10,000
105 = 100,000

24. Expanded Notation with Base-10 Exponents
When we expand using base-10 exponents, we just write the number as the sum (+) of each place’s base-10 exponent products:
64 = (6 x 101) + ( 4 x 100 )
643 = (6 x 102) + ( 4 x 101 ) + (3 x 100)

25. Expanded Notation with Base-10 Exponents: Your Turn!
Expand each number by its base-10 exponents. 72
193
5,408
62,937

26. Expanded Notation with Base-10 Exponents: Your Turn!
ANSWERS:
1) 72 =
(7 x 101) + (2 x 100)
2) 193 =
(1 x 102) + (9 x 101) + (3 x 100)
3) 5,408 =
(5 x 103) + (4 x 102) + (8 x 100)
4) 62,937 =
(6 x 104) + (2 x 103) + (9 x 102) +
(3 x 101) + (7 x 100)

27. POP QUIZ! Expand each number 3 ways: 461 5,387 90,256
a) by its place value: ;
b) by its expanded notation with base-10
products: (5 x 10) + (4 x 1); and
c) by its expanded notation with base-10
exponent products: (5 x 101) + (4 x 100).
461
5,387
90,256

28. POP QUIZ! ANSWERS: 1) 461 = a) 400 + 60 + 1
b) (4 x 100) + (6 x 10) + (1 x 1)
c) (4 x 102) + (6 x 101) + (1 x 100)
2) 5,387 =
a) 5,
b) (5 x 1,000) + (3 x 100) + (8 x 10) + (7 x 1)
c) (5 x 103) + (3 x 102) + (8 x 101) + (7 x 100)
3) 90,256 =
a) 90,
b) (9 x 10,000) + (2 x 100) + (5 x 10) + (6 x 1)
c) (9 x 104) + (2 x 102) + (5 x 101) + (6 x 100)