1. Place Value Basics: Decimals
for Dixon Elementary School’s
5th-Grade Math Classes

2. What Is a Decimal?
Decimals are numbers LESS THAN one whole. They are simply FRACTIONS of one whole.
Decimals are places that appear to the RIGHT of the ONES PLACE.
A DECIMAL POINT (.) separates whole numbers from decimals.

3. What Is a Decimal? Let’s look at the number: 123.456
The digits 123 are in the ONES PERIOD.
The digits 456 are in DECIMAL PLACES.

4. What Is a Decimal?
Let’s run over to CoolMath.com
Click here!

5. What Is a Decimal?
Just like with whole numbers, decimals are expressed in STANDARD, WORD, and EXPANDED forms (and notations).

6. Standard Form In STANDARD form, decimals are expressed with numbers:
0.1
0.12
0.123
0.1234

7. Word Form
In WORD form, decimals are expressed with the same terms we use in FRACTIONS:
0.1 = one-tenth = 1/10
0.01 = one hundredth = 1/100
0.001 = one thousandth = 1/1000

8. Word Form
To properly put decimals in word form, we first need to understand how to READ and SAY them.
Click here for CoolMath!
Click here for a Math Coach video!

9. Standard & Word Forms
So, we need to be able to go between expressing decimals in STANDARD and WORD forms, but we also want to express them as FRACTIONS:
0.6 = six-tenths = 6/10
0.35 = thirty-five hundredths = 35/100
2.572 = two AND five hundred seventy-two thousandths =
2-572/1000

10. Standard & Word Forms The HYPHEN rules are tricky.
Use a hyphen when expressing all TENTHS (four-tenths, eight-tenths, etc.)
Only use hyphens for two-digit numbers between with hundredths or thousandths (thirty-four hundredths, sixty-five thousandths, eight hundredths).

11. Standard & Word Forms: Your Turn
Write each in 3 forms: standard, word, and fractional forms.
Example: 2.5 = two and five-tenths = 2-5/10
0.3
1.43
sixteen hundredths
five and eighty-one thousandths
2/100
16/1000

12. Standard & Word Forms: Your Turn
ANSWERS:
0.3 = three-tenths = 3/10
1.43 = one and forty-three hundredths = 1-43/100
sixteen hundredths = 0.16 = 16/100
five and eighty-one thousandths = = 5-81/1000
2/100 = two hundredths = 0.02
16/1000 = sixteen thousandths = 0.016

13. Tomorrow…
…we EXPAND!

14. (EXPANDED NOTATION with EXPONENTS).
Expanding 3 Ways
We will learn to expand decimals in the same 3 ways we expanded whole numbers:
EXPANDED FORM,
EXPANDED NOTATION, and
SCIENTIFIC NOTATION
(EXPANDED NOTATION with EXPONENTS).

15. Expanded Form
When we EXPANDED whole numbers, we simply found the SUM (+) of the values of a number’s digits:
567 =

16. Expanded Form
We do the same when we EXPAND decimals. We simply found the SUM (+) of the values of a number’s digits:
5.67 =
-OR-
5 + 6/10 + 7/100

17. Expanded Form Let’s look at another one: 324.167 324.167 =
-OR-
/10 + 6/ /1000

18. Expanded Form Here’s one more: 609.208 609.208 = 600 + 9 + 0.2 + 0.08
/10 + 8/1000

19. Expanded Form: Your Turn
Let’s practice. Write the expanded form of each using DECIMALS and FRACTIONS:
Example: = and 9 + 3/10 + 5/100
4.6
32.89
6.407
18.003

20. Expanded Form: Your Turn
ANSWERS:
1) 4.6 =
4 + 6/10
2) =
/10 + 9/100
3) =
/10 + 9/ /1000
4) =
6 + 4/10 + 7/1000
5) =
/1000

21. Expanded Notation
In EXPANDED NOTATION, we found the SUM (+) of each digit multiplied by its place:
684 =
(6 x 100) + (8 x 10) + (4 x 1)

22. Expanded Notation In EXPANDED NOTATION with decimals, we do the same:
6.84 =
(6 x 1) + (8 x 0.1) + (4 x 0.01)
-OR-
(6 x 1) + (8 x 1/10) + (4 x 1/100)

23. (2 x 10) + (8 x 1) + (7 x 1/10) + (6 x 1/100) + (3 x 1/1000)
Expanded Notation
Here’s another example: =
(2 x 10) + (8 x 1) + (7 x 0.1) + (6 x 0.01) + (3 x 0.001)
-OR-
(2 x 10) + (8 x 1) + (7 x 1/10) + (6 x 1/100) + (3 x 1/1000)

24. Expanded Notation Let’s look at one more: 502.096 502.096 = -OR-
(5 x 100) + (2 x 1) + (9 x 0.01) + (6 x 0.001)
-OR-
(5 x 100) + (2 x 1) + (9 x 1/100) + (6 x 1/1000)

25. Expanded Notation: Your Turn
Write each number in EXPANDED NOTATION using DECIMALS and FRACTIONS.
Example: = (4 x 1) + (2 x 0.1) + (3 x 0.01)
(4 x 1) + (2 x 1/10) + (3 x 1/100)
9.2
17.45
8.947
29.307

26. Expanded Notation: Your Turn
ANSWERS:
1) 9.2 = (9 x 1) + (2 x 0.2)
(9 x 1) + (2 x 1/10)
2) = (1 x 10) + (7 x 1) + (4 x 0.1) + (5 x 0.01)
(1 x 10) + (7 x 1) + (4 x 1/10) + (5 x 1/00)
3) = (8 x 1) + (9 x 0.1) + (4 x 0.01) + (7 x 0.001)
(8 x 1) + (9 x 1/10) + (4 x 1/100) + (7 x 1/1000)
4) = (6 x 100) + (3 x 1) + (2 x 0.01)
(6 x 100) + (3 x 1) + (2 x 1/100)
5) = (2 x 10) + (9 x 1) + (3 x 0.1) + (7 x 0.001)
(2 x 10) + (9 x 1) + (3 x 1/10) + (7 x 1/1000)

27. Scientific Notation
SCIENTIFIC NOTATION expands numbers just like expanded notation, but we use base-10 exponents to identify each digit’s place:
268 =
(2 x 102) + (6 x 101) + (8 x 100)

28. Scientific Notation
In SCIENTIFIC NOTATION with decimals, we will learn two new concepts: 1) NEGATIVE exponents and 2) FRACTIONAL exponents:
2.68 =
(2 x 100) + (6 x 10-1) + (8 x 10-2)
-OR-
(2 x 100) + (6 x 1/101) + (8 x 1/1002)

29. Scientific Notation Holy smokes! Let’s look at another: 31.57 =
(3 x 101) + (1 x 100) + (5 x 10-1) + (7 x 10-2)
-OR-
(3 x 101) + (1 x 100) + (5 x 1/101) + (7 x 1/102)

30. Scientific Notation Here’s one more: 9.547 =
(9 x 100) + (5 x 10-1) + (4 x 10-2) + (7 x 10-3)
-OR-
(9 x 100) + (5 x 1/101) + (4 x 1/102) + (7 x 1/103)

31. Scientific Notation: Your Turn
Let’s practice. Use SCIENTIFIC NOTATION to expand each number using base-10 exponents and fractional exponents.
EXAMPLE: = (7 x 100) + (5 x 10-1) + (8 x 10-2)
(7 x 100) + (5 x 1/101) + (8 x 1/102)
4.23
9.675
32.04
2.605

32. Scientific Notation: Your Turn
ANSWERS:
1) 4.23 = (4 x 100) + (2 x 10-1) + (3 x 10-2)
(4 x 100) + (2 x 1/101) + (3 x 102)
2) = (9 x 100) + (6 x 10-1) + (7 x 10-2) + (5 x 10-3)
(9 x 100) + (6 x 1/101) + (7 x 1/102) + (5 x 1/103)
3) = (3 x 101) + (2 x 100) + (4 x 10-2)
(3 x 101) + (2 x 100) + (4 x 1/102)
4) = (2 x 100) + (6 x 10-1) + (5 x 10-3)
(2 x 100) + (6 x 1/101) + (5 x 1/103)

33. Whew! You did it! COMPARING decimals!
Coming up next…
COMPARING
decimals!

34. Comparing Decimals
Like we did with whole numbers, we express inequalities (numbers that are NOT equal) with these symbols:
< (less than)
> (greater than)
≠ (not equal to)
But when we move into decimals and fractions, this symbol is important, too: = (equal to).

35. Equivalent Whole Numbers
With whole numbers, there is not any confusion about numbers that are equal:
8 = 8
37 = 37
246 = 246
4,693 = 4,693
20,762 = 20,762

36. Equivalent Decimals With decimals, it isn’t quite so simple because:
.8 = 0.8
3.7 = 3.70
2.46 =
4 tenths = 40 hundredths
and
4 tenths = 400 thousandths

37. Equivalent Decimals
Just remember: any number of 0s to the right of the last non-0 number in a decimal is equivalent to the same decimal without the ending 0s:
0.72 = = =
1.6 = 1.60 = =
84.34 = = =

38. Equivalent Decimals
To dig a little deeper, let’s watch a TurtleDiary video on equivalent decimals…

39. Equivalent Decimals Click HERE to begin! Let’s practice a few on IXL.
When I call your team number, line up at the SmartBoard in the same order as your seat number (1-4).
When it’s your turn, choose the correct answer, click the submit button, and return to your seat.
Click HERE to begin!

40. Comparing Decimals
When we compare decimals, we use the same process we used to compare whole numbers: we start with the greatest place value and compare numbers in the same places moving to the right.
When comparing decimals, however, it’s easier to do when the numbers we compare have the same number of digits before and after the decimal.

41. Comparing Decimals
For example, if we want to compare12.2 and , it will be easier if we stack them, line up the decimals, and even up the places by adding 0s to empty spaces.
At first glance, it might seem that is greater because it has more digits, but once we even up the number of digits, we can see more clearly that 12.2 is greater:

42. Comparing Decimals Let’s watch a few videos: Video #1 Video #2

43. Comparing Decimals Click HERE to begin! Let’s practice a few on IXL.
When I call your team number, line up at the SmartBoard in the same order as your seat number (1-4).
When it’s your turn, choose the correct answer, click the submit button, and return to your seat.
Click HERE to begin!

44. Pop Quiz! Write a decimal equivalent to: 1.2 24.01 3) 965.207
Compare using <, = , or >.
4) 6.57 ___ 6.6
5) ___
6) ___ 65.04
7) ___
8) ___ .3

45. Pop Quiz ANSWERS! Write a decimal equivalent to:
1.2 = 1.20 or or …
24.01 = or …
3) = or …
Compare using <, = , or >.
4) 6.57 < 6.6
5) >
6) = 65.04
7) <
8) = .3

46. Great Job! ORDERING and ROUNDING decimals!
Coming up next…
ORDERING
and
ROUNDING
decimals!

47. Ordering Decimals
When we order decimals, just like with comparing them, it can be easier to do when we line up the decimals and even up the places by adding 0s.
But let’s watch Sal on KhanAcademy reason through ordering decimals WITHOUT doing so.

48. Ordering Decimals
Let’s practice. Order these decimals greatest to least:
Which number was greatest?
Which number was least?
Did you put them in this order?

49. Rounding Decimals
Let’s watch a MathAntics video to review rounding whole numbers and introduce us to rounding decimals.
And here’s one of Sal breaking down rounding decimals for us.

50. Rounding Decimals: Your Turn
Try rounding these numbers to the place in parentheses ( ).
1) (tenths)
Did you get 34.7?
2) (hundredths)
Did you get ?
3) (tenths)
Did you get 6?
4) (thousandths)
Did you get 10?

51. Pop Quiz Let’s practice comparing, ordering, and rounding! Compare:
9.4 ___ 9.398
68.07 ___
Order greatest to least:
Round to the given place:
4) (hundredths)
5) (tenths)

52. Pop Quiz: ANSWERS Let’s practice comparing, ordering, and rounding!
Compare:
9.4 > 9.398
68.07 =
Order greatest to least: 3)
Round to the given place:
4) (hundredths) = 85.68
5) (tenths) = 710