1. Who turned off the lights?
Factor Labeling
who
Who turned off the lights?
Factor Labeling
Factor Labeling
Factor Labeling
Factor Labeling
Factor Labeling
Factor Labeling
Richard J. Terwilliger by

2. Thanks, that sure is bright!
Factor Labeling
Thanks, that sure is bright!
Factor Labeling
Factor Labeling
Factor Labeling
Factor Labeling
Factor Labeling

3. Let’s review the method Factor Labeling Factor Labeling

4. we’ll determine the number of kilometers in 1 mile
Using the
we’ll determine the number
of kilometers in 1 mile
Factor Labeling Method
Factor Labeling
1 mi = ? km

5. To do this we need to know some basic conversions. 1 mile = 5280 feet
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

6. You should learn these common conversions. 1 mile = 5280 feet
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

7. To start the conversion from miles to kilometers
we draw a set of brackets.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

8. Then draw a horizontal line that divides the brackets, top to bottom.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

9. Again, we are converting
1 mi.
Again, we are converting
1 mile into kilometers
so we’ll place 1 mile
above the line.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

10. There’s no denominator so we’ll leave the bottom blank.
Leave this
blank
Leave this
blank
There’s no denominator
so we’ll leave the bottom
blank.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

11. We need to convert miles into feet,
feet into inches, inches into centimeters,
centimeters into meters and finally
meters into kilometers.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

12. Let’s get started and convert
1 mi.
Let’s get started and convert
the miles into feet!
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

13. Draw another set of brackets with a horizontal line next
1 mi.
Draw another set of brackets
with a horizontal line next
to the first set.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

14. We’re going to match units in the
1 mi.
mi.
We’re going to match units in the
numerator with units in the denominator.
Place miles in the bottom of the second
set of brackets. This matches the miles
already on the top in the first
set of brackets.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

15. 1 mi. ft. mi. We will change miles into feet so place feet
directly above miles
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

16. To fill in the conversion, place
1 mi.
5280
ft. 1
mi.
There’s 5280 feet in 1 mile.
To fill in the conversion, place
5280 next to feet and
1 next to miles.
Factor Labeling
conversions
1 mile = 5280 feet
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

17. As you can see, the miles on the
5280
ft. 1
mi.
As you can see, the miles on the
top of the first set of brackets matches
with the miles in the bottom of
the second set of brackets.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

18. so we have changed to feet.
1 mi.
mi.
5280
ft. 1
mi.
reduces to one
so we have changed to feet.
mi. 1
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

19. Watch closely as it’s done.
1 mi.
mi.
5280
ft. 1
mi.
The same process will
be used to change
feet into inches!
Watch closely as it’s done.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

20. 1 mi. mi. 5280 ft. 1 mi. Draw a set of brackets and then
the horizontal line.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

21. Match units top to bottom.
1 mi.
mi.
5280
ft. 1
mi.
ft.
Match units top to bottom.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

22. Place in the conversion units. (What are we converting to?
1 mi.
mi.
5280
ft.
in. 1
mi.
ft.
Place in the conversion units.
(What are we converting to?
In this case feet into inches.)
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

23. Fill in the conversion numbers.
1 mi.
mi.
5280
ft. 12
in. 1
mi. 1
ft.
Fill in the conversion numbers.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

24. Cross out the units that reduce.
1 mi.
mi.
5280
ft. 12
in. 1
mi. 1
ft.
Cross out the units that reduce.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

25. 1 mi. mi. 5280 ft. ft. 12 in. 1 mi. 1 ft. Again reduces to
leaving inches.
ft. 1
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

26. The next step is to convert inches into centimeters.
1 mi.
mi.
5280
ft.
ft. 12
in. 1
mi. 1
ft.
The next step is to convert
inches into centimeters.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

27. 1 mi. mi. 5280 ft. ft. 12 in. in. 2.54 cm. 1 mi. 1 ft. 1 in.
Follow the steps as
I demonstrate!.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

28. centimeters into meters.
1 mi.
mi.
5280
ft.
ft. 12
in.
in.
2.54
cm.
cm. 1 m. 1
mi. 1
ft. 1
in.
100
cm.
Now convert
centimeters into meters.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

29. The final step, converting meters into kilometers.
1 mi.
mi.
5280
ft.
ft. 12
in.
in.
2.54
cm.
cm. 1 m. m. 1
km. 1
mi. 1
ft. 1
in.
100
cm.
1000 m.
The final step, converting
meters into kilometers.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters
1 kilometer = 1000 meters

30. 1 mi. mi. 5280 ft. ft. 12 in. in. 2.54 cm. cm. 1 m. m. 1 km. 1 mi. 1
100
cm.
1000 m.
Our units are now in
kilometers!
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

31. To finish up we multiply
1 mi.
mi.
5280
ft.
ft. 12
in.
in.
2.54
cm.
cm. 1 m. m. 1
km. 1
mi. 1
ft. 1
in.
100
cm.
1000 m.
To finish up we multiply
all of the numbers
on the top.
(1 x 5280 x 12 x 2.54 x 1 x 1)
160934 =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

32. 1 mi. mi. 5280 ft. ft. 12 in. in. 2.54 cm. cm. 1 m. m. 1 km. 1 mi. 1
100
cm.
1000 m.
Multiply all of the
numbers on the bottom
and then divide.
(1 x 1 x 1 x 100 x 1000 )
160934 =
100000 =
1.61 km.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

33. 1 mi. mi. 5280 ft. ft. 12 in. in. 2.54 cm. cm. 1 m. m. 1 km. 1 mi. 1
100
cm.
1000 m.
1 mile = 1.61 kilometers
160934 =
100000 =
1.61 km.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 mi = ? km
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

35. complicated conversion?
Ready to do a more
complicated conversion?
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

36. 70 km ?m = hr s OK, Lets convert 70 kilometers/hour to meters/second?
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

37. 70 km ?m = hr s We start the conversion the same
way as before. Draw a set of
brackets and a horizontal line.
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

38. 70 km ?m = hr s 70 km. hr. Next, fill in the 70 kilometers per hour.
Kilometers on the top,
hours on the bottom.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

39. 70 km ?m = hr s 70 km. hr. For right now ignore the hours and convert
kilometers to meters.
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

40. 70 km ?m = hr s 70 km. hr. To change km to m,
draw another set of brackets
with the horizontal line.
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

41. 70 km ?m = hr s 70 km. hr. km. Fill in your units… km on the bottom to
match the km on the top .
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

42. And then meters on the top.70
km. m.
hr.
km.
And then meters on the top.
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

43. 70 km ?m = hr s 70 km. 1000 m. hr. 1 km. There are 1000 meters in
1 kilometer.
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters
1 kilometer = 1000 meters

44. 70 km ?m = hr s 70 km. 1000 m. hr. 1 km. The reduce and drop out. km.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

45. 70 km ?m = hr s 70 km. 1000 m. hr. 1 km. We now have meters per hour
and have to change
hours into seconds.
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

46. 70 km ?m = hr s 70 km. 1000 m. hr. 1 km. Another bracket set and
horizontal line.
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

47. 70
km.
1000 m.
hr.
hr. 1
km.
min.
Place hours on top and
minutes on the bottom.
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

48. 70
km.
1000 m. 1
hr.
hr. 1
km. 60
min.
There are 60 minutes
in 1 hour!
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

49. 70
km.
1000 m. 1
hr.
hr. 1
km. 60
min.
The hours reduce to one
and drop out.
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

50. 70 km ?m = hr s 70 km. 1000 m. 1 hr. hr. 1 km. 60 min. Now, convert
minutes into seconds!
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

51. 70 km ?m = hr s 70 km. 1000 m. 1 hr. 1 min. hr. 1 km. 60 min. 60 sec.
By now you should
have the hang of it so,
watch me do it!
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

52. 70 km ?m = hr s 70 km. 1000 m. 1 hr. 1 min. hr. 1 km. 60 min. 60 sec.
We now have meters
on the top and
seconds on the bottom!
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

53. 70 km ?m = hr s 70 km. 1000 m. 1 hr. 1 min. hr. 1 km. 60 min. 60 sec.
All that is left is to...
70 km ?m
hr s =
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

54. And then divide the bottom into the top70
km.
1000 m. 1
hr. 1
min.
70000 m.
19.4 m.
sec.
hr. 1
km. 60
min. 60
sec.
sec.
3600
Multiply the top,
70 km ?m
hr s =
Multiply the bottom,
And then divide the bottom into the top
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

55. 70 km ?m Whew ! = hr s 70 km. 1000 m. 1 hr. 1 min. 70000 m. 19.4 m.
sec.
hr. 1
km. 60
min. 60
sec.
sec.
3600
70 km ?m
hr s =
Whew !
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

57. Now let’s do a real tough one!, 1 mile = 5280 feet 1 foot = 12 inches
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

58. How about changing 1 mile = 5280 feet 1 foot = 12 inches
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

59. We’re going to solve this exactly the same way as the 2 m
min 3
We’re going to solve this
exactly the same way as the
other examples and for now we’ll
IGNORE the exponents.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

60. 2 m min 3 100 cm 1 m First change meters into centimeters.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

61. and between the numbers2 m
min 3
SPACE
100 cm
SPACE
SPACE 1
SPACE m
Notice the SPACE
I left after the units
and between the numbers
and the units!
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

62. to be used for the exponents.2 m
min 3
100 cm 1 m
This SPACE is going
to be used for the exponents.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

63. Since the meters is squared in the first set of brackets, 2 2 m
100 cm 3 1 2
min m
Since the meters is squared
in the first set of brackets,
we will square the meters
in the second set.
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

64. square everything else!2 2 2 2 m
100 cm 3 2 1 2
min m
Now,
If we square the meters,
we MUST
square everything else!
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

65. We’ll do the same thing againcm 1
100 2 2 2 m 1
min 3
min 60
sec
We’ll do the same thing again
changing
into 3
min
sec
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

66. Remember to leave SPACEcm 1
100 2 2 2 m 1
min 3
min 60
sec
Remember to leave SPACE
for the
exponents!
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

67. We’re changing from minutes cubed so everything in the third bracket cm 1
100 2 3 3 2 2 m 1
min 3 3 3
min 60
sec
We’re changing from minutes cubed so
everything in the third bracket
is cubed
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

68. m cm 1 100 2 3 3 2 2 m 1 min 3 3 3 min 60 sec Lets REDUCE
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

69. m cm 1
100 2 m cm 1
100 2 3 3 2 2 m 1
min 3 3 3
min 60
sec
And we’re left with…. cm
sec 2 3
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

70. To finish up, multiply the top and divide by the bottom!cm 1
100 2 m cm 1
100 2 3 3 2 2 m 1
min 3 3 3
min 60
sec
To finish up, multiply the top
and divide by the bottom!
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

71. m cm 1
100 2 m cm 1
100 2 3 3 2
.09 cm
sec 2 3 2 m 1
min 3 3 3
min 60
sec
2 x 1002
603
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

72. m cm 1
100 2 m cm 1
100 2 3 3 2 2 2 m 1
min cm
.09 3 3 3 3
min 60
sec
sec
Now, wasn’t that easy!
Factor Labeling
conversions
1 mile = 5280 feet
1 foot = 12 inches
1 inch = 2.54 centimeters
1 meter = 100 centimeters
1 kilometer = 1000 meters

73. Factoring Labeling by
Richard J. Terwilliger
Suffern High School

74. Hey, Who turned out the lights?