Lesson
5-4 Length in the Metric System
KEY Concept
VOCABULARY
The metric system is used throughout the world. It is based on
powers of ten. The most commonly used units are millimeters,
centimeters, meters, and kilometers. The base unit of length in
the metric system is the meter .
metric system
a measurement system
based on powers of 10
that includes units such
as meter, gram, and liter
Unit
Abbreviation
Number of Meters
millimeter
mm
0.001 m
centimeter
cm
0.01 m
decimeter
dm
0.1 m
meter
m
1m
dekameter
dkm
10 m
hectometer
hm
100 m
kilometer
km
1,000 m
÷
length
a measurement of the
distance between two
points
×
To convert between units, multiply or divide by powers of ten.
When converting from bigger to smaller units, multiply.
When converting from smaller to larger units, divide.
A centimeter is one unit smaller than a decimeter.
5.0 dm = __ cm
5
km
hm
dkm
m
dm
cm
5
0
mm
1 dm = 10 cm
Multiplying by 10
moves the decimal
point 1 place to
the right.
5 · 10 = 50 cm
A kilometer is two units larger than a dekameter.
5.0 dkm = __ km
5
km
hm
dkm
0
0
5
m
dm
5 ÷ 102 = 5 ÷ 100 = 0.05 km
cm
mm
meter
a base unit in the metric
system for measuring
length
1 km = 100 dkm
Dividing by 100
moves the decimal
point 2 places to
the left.
Example 1
YOUR TURN!
Convert.
Convert.
8.2 m = ____ mm
1. Are you converting to a smaller unit or a
larger unit?
1.3 km = ____ cm
1. Are you converting to a smaller unit or a
larger unit?
smaller
2. Do you multiply or divide?
2. Do you multiply or divide?
multiply
3. How many units are you converting?
3. How many units are you converting?
three
4. Use
4. Use 103.
8.2 · 103 =
8.2 · 1,000 = 8,200
8.2 m = 8,200 mm
.
1.3
10
1.3
=
1.3 km =
Example 2
YOUR TURN!
Convert.
Convert.
613.5 mm =____ dkm
1. Are you converting to a smaller unit or
a larger unit?
=
2,234 m = ____ km
1. Are you converting to a smaller unit
or a larger unit?
larger
2. Do you multiply or divide?
2. Do you multiply or divide?
divide
3. How many units are you converting?
3. How many units are you converting?
four
4. Use 104.
613.5 ÷ 104 =
613.5 ÷ 10,000 = 0.06135
613.5 mm = 0.06135 dkm
4. Use
2,234
2,234
2,234 m =
.
10
=
=
Guided Practice
Convert each measurement.
1
650 mm = ____ dm
2
units
A decimeter is
than a millimeter.
650
3
10
A dekameter is
than a centimeter.
280,900
=
3,467 km = ____ m
4
units
A meter is
than a kilometer.
3,467
10
280,900 cm = ____ dkm
10
units
=
7,431 m = ____ dm
A decimeter is
than a meter.
7,431
=
10
unit
=
Step by Step Practice
5
Convert.
80,743 dm = ____ km
Step 1 Are you converting to a smaller unit or a larger unit?
Step 2 Do you multiply or divide?
Step 3 How many units are you converting?
Step 4 Use
.
10
80,743
= 80,743
=
Step 5 80,743 dm =
Convert each measurement.
6
10.75 km = ____ hm
7
405 dm = ____ hm
by 10 .
10.75
10
=
by 10
405
10
=
.
Convert each measurement.
8
14.4 m = ____ mm
10
14.4
10
9
325
=
6.5 dm = ____ km
6.5
10
325 hm = ____ m
11
10
742 mm = ____ cm
742
=
=
10
=
Step by Step Problem-Solving Practice
Solve.
12
SWIMMING Reece swims 50 meters in a swim meet. How
many hectometers did she swim?
units
Hectometers are
by 10
than meters.
.
place(s) to the
The decimal moves
.
Check off each step.
Understand: I underlined key words.
Plan: To solve the problem, I will
.
Solve: The answer is
.
Check: I checked my answer by
.
Skills, Concepts, and Problem Solving
Convert each measurement.
13
45 dm =
mm
14
925 dm =
hm
Convert each measurement.
15
5,487 m =
17
79 km =
19
631 mm =
cm
16
36,725 mm =
hm
18
489 dkm =
m
20
568,734 dm =
dkm
dm
km
Solve.
21
RUNNING Emma ran 11 laps around the track during practice
one day. How many kilometers did Emma run during practice?
22
SCHOOL SUPPLIES Ethan has new pencils that are each 140 mm
long. How long is each pencil in centimeters?
23
DRIVING Garrett drove 62 kilometers to see his grandmother. How
many dekameters did he drive?
1 lap = 400m
Vocabulary Check Write the vocabulary word that completes each sentence.
24
The
25
The distance between two points is called the
26
The measuring system based on powers of 10 is the
27
is the base unit for measuring length in the metric system.
.
Write in your own words how to convert between metric
units. Do you think it is easier to convert in the metric
system or the customary system? Explain your answer.
.
Chapter
5
Progress Check 2
(Lessons 5-3 and 5-4)
Convert each measurement.
1
80 oz =
3
1.5 lbs =
5
5,000 cm =
7
10 m =
9
200 dkm =
lb
oz
m
cm
cm
2
2T=
lb
4
1T =
oz
6
2 km =
8
150 mm =
10
2,500 mm =
Solve.
11
ANIMALS In a report on giraffes, Sumintra wrote that one
animal weighed 1.5. She didn’t write down the units. What
unit would be appropriate for the weight of a giraffe? How
many pounds does the giraffe weigh?
12
DISTANCE Aisha lives 2.5 km from her friend Jason. When she
walks to his house, how many meters does she have to walk?
13
MODEL CARS Jacob and Carlos had a contest to see how far their
model cars would go on one wind up. Jacob’s car went 1.6 m.
Carlos’ car went 145 cm. Whose car went farther? By how much?
dkm
hm
m
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Units, Dimensions and Measurement 1
Notes and exercises
by
Pradeep Kshetrapal
P
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9Tmt IndirectProported
1.1 Physical Quantity.
A quantity which can be measured and by which various physical happenings can be explained and
expressed in form of laws is called a physical quantity. For example length, mass, time, force etc.
On the other hand various happenings in life e.g., happiness, sorrow etc. are not physical quantities
because these can not be measured.
Measurement is necessary to determine magnitude of a physical quantity, to compare two similar
physical quantities and to prove physical laws or equations.
A physical quantity is represented completely by its magnitude and unit. For example, 10 metre means
a length which is ten times the unit of length 1 kg. Here 10 represents the numerical value of the given
quantity and metre represents the unit of quantity under consideration. Thus in expressing a physical
quantity we choose a unit and then find that how many times that unit is contained in the given physical
quantity, i.e.
anxut
Physical quantity (Q) = Magnitude × Unit = n × u
NIU
Where, n represents the numerical value and u represents the unit. Thus while expressing definite
amount of physical quantity, it is clear that as the unit(u) changes, the magnitude(n) will also change but
product ‘nu’ will remain same.
Notationof Inverse Proportion
i.e.
n u = constant,
or
n1u1
n2u 2
constant ;
n
1
u
i.e. magnitude of a physical quantity and units are inversely proportional to each other .Larger the
unit, smaller will be the magnitude.
1.2 Types of Physical Quantity.
(1) Ratio (numerical value only) : When a physical quantity is a ratio of two similar quantities, it has
no unit.
e.g. Relative density = Density of object/Density of water at 4oC
Refractive index = Velocity of light in air/Velocity of light in medium
Strain = Change in dimension/Original dimension
Note :
Angle is exceptional physical quantity, which though is a ratio of two similar physical
quantities (angle = arc / radius) but still requires a unit (degrees or radians) to specify it along with
its numerical value.
K
(2) Scalar (Magnitude only) : These quantities do not have any direction e.g. Length, time, work,
energy etc.
Magnitude of a physical quantity can be negative. In that case negative sign indicates that the
numerical value of the quantity under consideration is negative. It does not specify the direction.
Scalar quantities can be added or subtracted with the help of following ordinary laws of addition or
subtraction.
Page
Vector physical quantities can be added or subtracted according to vector laws of addition. These laws
are different from laws of ordinary addition.
7
(3) Vector (magnitude and direction) : e.g. displacement, velocity, acceleration, force etc.
Note :
11ft
magnitude
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Units, Dimensions and Measurement 8
vector
magnitude with direction
There are certain physical quantities which behave neither as scalar nor as vector. For
example, moment of inertia is not a vector as by changing the sense of rotation its value is
not changed. It is also not a scalar as it has different values in different directions (i.e. about
different axes). Such physical quantities are called Tensors.
1.3 Fundamental and Derived Quantities.
(1) Fundamental quantities : A few physical quantities which are independent of all other quantities
and do not require the help of any other physical quantity for their definition, These quantities are also
called fundamental or base quantities.
Maes
etc
Egg censer
Time
(2) Derived quantities : All other physical quantities can be derived by suitable multiplication or
division of different powers of fundamental quantities. These are therefore called derived quantities.
If length is defined as a fundamental quantity
Area and volume are derived from length
and are expressed in term of length with power 2 and 3 over the term of length.
1.4 Fundamental and Derived Units.
System of units : A complete set of units, both fundamental and derived for all kinds of physical
quantities is called system of units. The common systems are given below –
second
centimeter gram
system
(1) CGS system : The system is also called Gaussian system of units. In it length, mass and time have
been chosen as the fundamental quantities .
corresponding fundamental units are centimetre (cm), gram (g) and second (s) respectively.
Jeterkilogram
seÉÉ
(2) MKS system : The system is also called Giorgi system. In this system also length, mass and time
have been taken as fundamental quantities,
corresponding fundamental units are metre, kilogram and second.
Foot pound system
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8
(3) FPS system : In this system foot, pound and second are used respectively for measurements of
length, mass and time. In this system force is a derived quantity with unit poundal.
genius PHYSICS
by Pradeep Kshetrapal
Units, Dimensions and Measurement 9
(4) S. I. system : It is known as International system of units, and is infact extended system of units
applied to whole physics. There are seven fundamental quantities in this system. These quantities and their
units are given in the following table
Quantity
Name of Unit
Symbol
Length
metre
m
Mass
kilogram
kg
Time
second
s
Electric Current
ampere
A
Temperature
Kelvin
K
Amount of Substance
mole
mol
Luminous Intensity
candela
cd
Besides the above seven fundamental units two supplementary units are also defined –
Radian (rad) for plane angle and Steradian (sr) for solid angle.
Note :
Apart from fundamental and derived units we also use very frequently practical units.
These may be fundamental or derived units
e.g., light year is a practical unit (fundamental) of distance while horse power is a practical unit
(derived) of power.
Practical units may or may not belong to a system but can be expressed in any system of units
e.g., 1 mile = 1.6 km = 1.6 × 103 m.
1.5 S.I. Prefixes.
Prefix
Symbol
1018
exa
E
1015
peta
P
1012
tera
T
109
giga
G
106
mega
M
103
kilo
k
102
hecto
h
101
deca
da
10–1
deci
d
10–1
centi
c
10–3
milli
m
10–6
micro
Page
Power of 10
9
In physics we have to deal from very small (micro) to very large (macro) magnitudes as one side we
talk about the atom while on the other side of universe, e.g., the mass of an electron is 9.1 10–31 kg while
that of the sun is 2 1030 kg. To express such large or small magnitudes simultaneously we use the
following prefixes :
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by Pradeep Kshetrapal
Units, Dimensions and Measurement 10
10–9
nano
n
10–12
pico
p
10–15
femto
f
10–18
atto
a
/ from very small (micro) to very large (macro) magnitudes
1.7 Practical Units.
(1) Length :
(i) 1 fermi = 1 fm = 10–15 m
(ii) 1 X-ray unit = 1XU = 10–13 m
(iii) 1 angstrom = 1Å =
10–10
m=
Micro
10–8
cm =
10–7
mm = 0.1 mm
(iv) 1 micron = m = 10–6 m
(v) 1 astronomical unit = 1 A.U. = 1. 49
(vi) 1 Light year = 1 ly = 9.46
1011 m
1.5
1011 m
108 km
Macro
1015 m
(vii) 1 Parsec = 1pc = 3.26 light year
(2) Mass :
(i) Chandra Shekhar unit : 1 CSU = 1.4 times the mass of sun = 2.8
1030 kg
(ii) Metric tonne : 1 Metric tonne = 1000 kg
(iii) Quintal : 1 Quintal = 100 kg
(iv) Atomic mass unit (amu) : amu = 1.67
10–27 kg mass of proton or neutron is of the order of 1 amu
(3) Time :
(i) Year : It is the time taken by earth to complete 1 revolution around the sun in its orbit.
(ii) Lunar month : It is the time taken by moon to complete 1 revolution around the earth in its orbit.
Page
(iii) Solar day : It is the time taken by earth to complete one rotation about its axis with respect to sun.
Since this time varies from day to day, average solar day is calculated by taking average of the duration of all
the days in a year and this is called Average Solar day.
10
1 L.M. = 27.3 days
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Units, Dimensions and Measurement 11
1 Solar year = 365.25 average solar day
or
average solar day
1
the part of solar year
365 .25
(iv) Sedrial day : It is the time taken by earth to complete one rotation about its axis with respect to a
distant star.
1 Solar year = 366.25 Sedrial day = 365.25 average solar day
Thus 1 Sedrial day is less than 1 solar day.
(v) Shake : It is an obsolete and practical unit of time.
1 Shake = 10– 8 sec
1.8 Dimensions of a Physical Quantity.
When a derived quantity is expressed in terms of fundamental quantities, it is written as a product of
different powers of the fundamental quantities. The powers to which fundamental quantities must be raised
in order to express the given physical quantity are called its dimensions.
To make it more clear, consider the physical quantity force
Force = mass × acceleration
mass velocity
time
mass length/time
= mass × length × (time)–2 .... (i)
time
Thus, the dimensions of force are 1 in mass, 1 in length and – 2 in time.
Here the physical quantity that is expressed in terms of the base quantities is enclosed in square
brackets to indicate that the equation is among the dimensions and not among the magnitudes.
Thus equation (i) can be written as [force] = [MLT–2].
Such an expression for a physical quantity in terms of the fundamental quantities is called the
dimensional equation. If we consider only the R.H.S. of the equation, the expression is termed as
dimensional formula.
Thus, dimensional formula for force is, [MLT – 2].
1.9 Important Dimensions of Complete Physics.
Mechanics
(1)
Velocity or speed (v)
m/s
[M0L1T –1]
(2)
Acceleration (a)
m/s2
[M0LT –2]
(3)
Momentum (P)
kg-m/s
[M1L1T –1]
(4)
Impulse (I)
Newton-sec or kg-m/s
[M1L1T –1]
(5)
Force (F)
Newton
[M1L1T –2]
(6)
Pressure (P)
Pascal
[M1L–1T –2]
(7)
Kinetic energy (EK)
Joule
[M1L2T –2]
(8)
Power (P)
Watt or Joule/s
[M1L2T –3]
(9)
Density (d)
kg/m3
[M1L– 3T 0]
(10)
Angular displacement ( )
Radian (rad.)
[M0L0T 0]
(11)
Angular velocity ( )
Radian/sec
[M0L0T – 1]
(12)
Angular acceleration ( )
Radian/sec2
[M0L0T – 2]
(13)
Moment of inertia (I)
kg-m2
Unit
Dimension
[M1L2T0]
11
Quantity
Page
S. N.
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by Pradeep Kshetrapal
Units, Dimensions and Measurement 12
S. N.
Quantity
(14)
Torque ( )
Newton-meter
[M1L2T –2]
(15)
Angular momentum (L)
Joule-sec
[M1L2T –1]
(16)
Force constant or spring constant (k)
Newton/m
[M1L0T –2]
(17)
Gravitational constant (G)
N-m2/kg2
[M–1L3T – 2]
(18)
Intensity of gravitational field (Eg)
N/kg
[M0L1T – 2]
(19)
Gravitational potential (Vg)
Joule/kg
[M0L2T – 2]
(20)
Surface tension (T)
N/m or Joule/m2
[M1L0T – 2]
(21)
Velocity gradient (Vg)
Second–1
[M0L0T – 1]
(22)
Coefficient of viscosity ( )
kg/m-s
[M1L– 1T – 1]
(23)
Stress
N/m2
[M1L– 1T – 2]
(24)
Strain
No unit
(25)
Modulus of elasticity (E)
N/m2
(26)
Poisson Ratio ( )
No unit
[M0L0T 0]
(27)
Time period (T)
Second
[M0L0T1]
(28)
Frequency (n)
Hz
Unit
Dimension
[M0L0T 0]
[M1L– 1T – 2]
[M0L0T –1]
Heat
S. N.
Quantity
Dimension
(1)
Temperature (T)
Kelvin
[M0L0T0
(2)
Heat (Q)
Joule
[ML2T– 2]
(3)
Specific Heat (c)
Joule/kg-K
[M0L2T– 2
–1]
(4)
Thermal capacity
Joule/K
[M1L2T – 2
–1]
(5)
Latent heat (L)
Joule/kg
[M0L2T – 2]
(6)
Gas constant (R)
Joule/mol-K
[M1L2T– 2
– 1]
(7)
Boltzmann constant (k)
Joule/K
[M1L2T– 2
– 1]
(8)
Coefficient of thermal conductivity (K)
Joule/m-s-K
[M1L1T– 3
– 1]
(9)
Stefan's constant ( )
Watt/m2-K4
[M1L0T– 3
– 4]
(10)
Wien's constant (b)
Meter-K
[M0L1To 1]
(11)
Planck's constant (h)
Joule-s
[M1L2T–1]
(12)
Coefficient of Linear Expansion ( )
Kelvin–1
[M0L0T0
(13)
Mechanical eq. of Heat (J)
Joule/Calorie
[M0L0T0]
(14)
Vander wall’s constant (a)
Newton-m4
[ML5T– 2]
(15)
Vander wall’s constant (b)
m3
[M0L3T0]
Unit
1]
–1]
Electricity
Unit
Dimension
(1)
Electric charge (q)
Coulomb
[M0L0T1A1]
(2)
Electric current (I)
Ampere
[M0L0T0A1]
(3)
Capacitance (C)
Coulomb/volt or Farad
[M–1L– 2T4A2]
(4)
Electric potential (V)
Joule/coulomb
M1L2T–3A–1
12
Quantity
Page
S. N.
5-5 Capacity in the Metric
System
KEY Concept
VOCABULARY
The base unit of capacity in the metric system is the liter .
capacity
the amount of dry or
liquid material a container
can hold
Unit
Symbol
Number of Liters
milliliter
mL
0.001 L
centiliter
cL
0.01 L
deciliter
dL
0.1 L
liter
L
1L
dekaliter
dkL
10 L
hectoliter
hL
100 L
kiloliter
kL
1,000 L
÷
liter
a base metric unit for
measuring capacity
metric system
a measurement system
that includes units such
as meter, gram, and liter
×
To convert between units, multiply or divide by powers of ten.
When converting from bigger to smaller units, multiply.
When converting from smaller to larger units, divide.
A milliliter is one unit smaller than a centiliter.
3.0 cL = __ mL
1 cL = 10 cL
3
kL
hL
dkL
L
dL
cL
mL
3
0
Multiplying by 10
moves the decimal
point 1 place to
the right.
3 · 10 = 30 mL
A hectoliter is three units larger than a deciliter.
7.0 dL = __ hL
7
kL
hL
dkL
L
dL
0
0
0
7
1 hL = 1,000 dL
cL
7 ÷ 103 = 7 ÷ 1,000 = 0.007 hL
mL
Dividing by 1,000
moves the decimal
point 3 places to
the left.
The most commonly used units are liter and milliliter. The steps for
converting between units of capacity are the same as the steps for
converting between units of length.
GO ON
Example 1
YOUR TURN!
Convert. 3.62 kL = ____ L
Convert.
1. Are you converting to a smaller unit or
a larger unit? smaller
1. Are you converting to a smaller unit or
2. Do you multiply or divide? multiply
2. Do you multiply or divide?
3. How many units are you converting?
three
3. How many units are you converting?
4. Use 103.
4. Use
61 L = ____ mL
a larger unit?
3.62 · 103 = 3.62 · 1,000
61
.
10
= 3,620
= 61
=
61 L =
3.62 kL = 3,620 L
Example 2
YOUR TURN!
Convert. 5,813 cL = ____ kL
Convert.
1. Are you converting to a smaller unit or
a larger unit? larger
1. Are you converting to a smaller unit or
2. Do you multiply or divide? divide
2. Do you multiply or divide?
3. How many units are you converting?
3. How many units are you converting?
360 mL = ____ dkL
a larger unit?
five
4. Use
4. Use 105.
360
5,813 ÷ 105 = 5,813 ÷ 100,000
.
10
= 360
=
= 0.05813
360 mL =
5,813 cL = 0.05813 kL
Guided Practice
Convert each measurement.
1
437 hL = ____ cL
2
unit(s)
A centiliter is
175 dkL = ____ L
A liter is
unit(s)
than a hectoliter.
437
10
=
than a dekaliter.
175
10
=
Step by Step Practice
3
Convert.
7,450 cL = ____ hL
Step 1 Are you converting to a smaller unit or a larger unit?
Step 2 Do you multiply or divide?
Step 3 How many units are you converting?
Step 4 Use
.
10
= 7,450
=
Step 5 7,450 cL =
Convert each measurement.
4
384 mL = ____ dL
by 10
384
10
5
24,500 dL = ____ kL
.
by 10 .
24,500
=
10
=
Step by Step Problem-Solving Practice
Solve.
6
SOCCER Camille’s soccer team drinks 18 liters of a sports
drink during a game. How many milliliters of sports drink
does the team drink during a game?
Milliliters are
unit(s)
by 10
than liters.
.
The decimal moves
place(s) to the
.
Check off each step.
Understand: I underlined key words.
Plan: To solve the problem, I will
.
Solve: The answer is
.
Check: I checked my answer by
.
GO ON
Skills, Concepts, and Problem Solving
Convert each measurement.
7
175 kL =
dkL
8
321 L =
mL
9
9.62 cL =
hL
10
86 hL =
L
11
5,000 kL =
L
12
4 mL =
cL
13
76,500 dkL =
14
6 hL =
mL
cL
Solve.
15
TRUCKING Brock is a truck driver. He used 2,500 liters
of gas on his last trip. How many kiloliters of gas did he use?
16
MEDICINE One dose of a children’s medicine is 5 mL. How
many centiliters are in one dose?
17
SWIMMING A school swimming pool holds 375,000 liters of water.
How many millimeters of water does it hold?
Vocabulary Check Write the vocabulary word that completes each sentence.
18
A(n)
19
is the amount of dry or liquid material that a
container can hold.
20
The measuring system that includes milliliters, liters, and kiloliters
is the
21
is the base metric unit for measuring capacity.
.
Compare and contrast the units of liter and meter.
Lesson
5-6 Mass in the Metric System
KEY Concept
VOCABULARY
Mass in the metric system is measured using the following
units. The base unit of mass in the metric system is the gram .
gram
a base metric unit for
measuring mass
Unit
Symbol
Number of Meters
milligram
mg
0.001 g
centigram
cg
0.01 g
decigram
dg
0.1 g
gram
g
1g
dekagram
dkg
10 g
hectogram
hg
100 g
kilogram
kg
1,000 g
÷
metric system
a measurement system
that includes units such
as meter, gram, and liter
×
To convert between units, multiply or divide by
powers of ten.
When converting from bigger to smaller units, multiply. . .
When converting from smaller to larger units, divide.
A gram is one unit smaller than a dekagram.
8.0 dkg = __ g
8
kg
hg
dkg
g
8
0
dg
cg
mg
1 dkg = 10 g
Multiplying by 10
moves the decimal
point 1 place to
the right.
8 · 10 = 80 g
A decigram is two units larger than a milligram.
2.0 mg = __ dg
2
kg
hg
dkg
g
dg
cg
mg
0
0
2
mass
the amount of matter
in an object
1 dg = 100 mg
Dividing by 100
moves the decimal
point 2 places to
the left.
2 ÷ 102 = 2 ÷ 100 = 0.02 dg
The most commonly used units for mass are the milligram, gram,
and kilogram.
Example 1
YOUR TURN!
Convert.
Convert.
1.7 kg = ____ cg
5,662 g = ____ mg
1. Are you converting to a smaller unit or
a larger unit? smaller
1. Are you converting to a smaller unit or
2. Do you multiply or divide? multiply
2. Do you multiply or divide?
3. How many units are you converting?
3. How many units are you converting?
a larger unit?
five
4. Use
4. Use 105.
.
5,662
1.7 · 105 =
10
=
5,662
1.7 · 100,000 = 170,000
=
5,662 g =
1.7 kg = 170,000 cg
Example 2
YOUR TURN!
Convert.
Convert.
483 cg = ____ dkg
3,601.4 mg = ____ g
1. Are you converting to a smaller unit or
a larger unit? larger
1. Are you converting to a smaller unit or
2. Do you multiply or divide? divide
2. Do you multiply or divide?
3. How many units are you converting?
3. How many units are you converting?
a larger unit?
three
4. Use
4. Use 103.
.
3,601.4
483 ÷ 103 =
10
483 ÷ 1,000 = 0.483
3,601.4
483 cg = 0.483 dkg
3,6014 mg =
=
=
Guided Practice
Convert each measurement.
1
4,324 cg = ____ g
A gram is
than a centigram.
4,324
10
=
2
units
6.8 dkg = ____ hg
A hectogram is
than a dekagram.
6.8
10
=
unit
Step by Step Practice
3
Convert.
345,000 g = ____ kg
Step 1 Are you converting to a smaller unit or a larger unit?
Step 2 Do you multiply or divide?
Step 3 How many units are you converting?
Step 4 Use
.
10
345,000
Step 5
= 345,000
=
345,000 g =
Convert each measurement.
4
640 cg = ____ dkg
5
3.4 dkg = ____ dg
by 10 .
640
10
by 10 .
3.4
=
10
=
Step by Step Problem-Solving Practice
Solve.
6
HEALTH A bottle of vitamins has 250 tablets. Each
tablet has 200 mg of Vitamin C in it. How many grams of
Vitamin C are in one bottle?
Find the total number of milligrams in the bottle.
200 mg ·
=
Convert milligrams to grams.
÷
=
Check off each step.
Understand: I underlined key words.
Plan: To solve the problem, I will
.
Solve: The answer is
.
Check: I checked my answer by
. GO ON
Step by Step Practice
3
Find the area of the figure.
3 in.
Step 1 Identify the shape.
10 in.
Step 2 Write the formula for the area.
Step 3 Name the value for each variable.
Step 4 Substitute the values into the formula.
Step 5 Simplify.
A=
(
)=
Find the area of each figure.
4 ft
4
5
8 mm
5 ft
3 ft
16 mm
8 ft
The figure is a
1 h(b + b )
A = __
1
2
2
.
The figure is a
.
A=
A=
A=
Step by Step Problem-Solving Practice
Solve.
6
ART PROJECT Brianne has to make a collage for art class on
a poster board that has dimensions 16 inches by 22 inches.
Because Brianne has to use the whole board, what is the total
area she needs to cover with her collage?
The board is a
.
A=
Check off each step.
Understand: I underlined key words.
Plan: To solve the problem, I will
.
Solve: The answer is
.
Check: I checked my answer by
.
GO ON
-To convert between units of area,
multiply or divide by squares of
powers of ten
When converting from bigger to smaller units,
multiply.
When converting from smaller to larger
units, divide.
To convert between units of volume,
multiply or divide by cube of powers of ten
When converting from bigger to smaller units, multiply. . .
When converting from smaller to larger units, divide.
Relation between Volume
and
Capacity
1ml equal to ?????Cm3
1000am
1000mL
I
1 ml = 1 g
Relation between volume and capacity
Measuring Density
What is density
1
How to measure density
2
Measuring matter in different systems
3
4
mass
9
Chapter
Progress Check 3
5
(Lessons 5-5, 5-6, and 5-7)
Convert each measurement.
1
300 cL =
L
2
6,000 mg =
3
4 kL =
liters
4
16 hL =
cL
5
12 g =
kg
6
11 hg =
g
7
150 kg =
8
19 dL =
hL
dg
g
Find the perimeter and the area of each figure.
9
10
15 m
16 ft
5 ft
3 ft
8 ft
25 m
11
12
30 in.
5 ft
32 in.
13 yd
85 yd
84 yd
36 in.
Solve.
13
GROCERIES Mehlia bought 6,804 g of fruit at the grocery.
How many kilograms of fruit did she buy?
14
HEALTH Doctors recommend that adults drink about 1,900 mL
of water every day. About how many liters of water should an
adult drink per day?
15
SCIENCE In an experiment, Reina measured 145 mL of
hydrochloric acid. How many liters did she have?
Chapter
Chapter Test
5
Convert each measurement.
1
1 km =
hm
1 hm =
dkm
1 hg = 10
1 hL =
1 dkm =
m
1 dkg = 10
1
1 g = 10
1 L = 10
1m=
4
2
dm
1 kg =10
1 kL =10
3
dkL
= 10
1 dm =
cm
1 dg = 10
1 dL =
1 cm =
mm
1 cg = 10
1
15 dkL =
kL
hL
mL
dkL
L
dL
cL
5
mL
= 10 mL
4,120 g =
kg
hg
cL
kg
dkg
g
dg
cg
6
192 in. =
ft
7
1.7 km =
cm
8
1,200 c =
gal
9
4T=
10
16 c =
fl oz
11
5 mi =
ft
12
72 oz =
lb
13
6.7 g =
mg
14
800 L =
cL
15
328 mm =
oz
m
mg
Which are the basic units in the metric system?
MAGNITUDE
BASIC UNIT
SYMBOL
LENGTH
metre
m
MASS
gram
g
CAPACITY
litre
L
VOLUME
cube metre
m3
SURFACE
square metre
m2
Note: you can write meter or metre, liter or litre; it depends if it is American or British English.
Work in groups to find objects you can measure with these units:
a) metre: a table, _________________________________________
b) gram: sugar pot, ____________________________________________
c) litre: oil, _______________________________________________
d) cube metre: a swimming pool, __________________________________
e) square metre: a room, ___________________________________________
For multiples and submultiples we use prefixes that multiply or divide the basic unit by
powers of 10
Meanings of metric prefixes:
1000 = 103
KiloHecto-
100 = 102
Deca- or (deka-)
10 = 101
Basic
Multiples
unit
Deci-
0,1 = 10–1
Centi-
0,01 = 10–2
Milli-
0,001 = 10–3
Submultiples
1. LENGTH Measurements of 1 dimension
How long is your pencil?
Measure it using a ruler from one end to another.
It is about 12 cm long.
Length is a measure of how long or wide something is.
For example: the bed is 1,90 m long and 80 cm wide.
What’s the perimeter of the bed? We have to add twice the length plus twice the width:
Length 1,90 x 2 = 3,80 m
Width 80 x 2 = 160 cm
To add length plus width we must change the amounts to the same units: m or cm
3,80 m = 380 cm + 160 cm = 540 cm or 160 cm = 1,60 m + 3,80 = 5,40 m
The basic unit of length is the metre. Multiples and submultiples are:
Kilometre
Hectometre Decametre metre
decimetre
1km=
1000m
1 hm =
100m
1dm=0,1m 1cm=
0,01m
1 dam=
10m
MultIples
centimetre millimetre
1mm=
0,001m
Submultiples
How many dm, cm and mm are in one metre? 1 m = 10 dm, 100 dm and 1000 mm
Exercise:
1.Choose the most appropriate unit – km, m, cm, mm – to measure:
a) a pen
b) a stamp c) a building d) the distance from London to Oxford e) an eraser
2. Choose the most reasonable measurement:
1. Width of your hand
a) 95 mm
2. Height of an adult
a) 163 mm
3. Length of a pair of scissors
b) 9,5 dm
c) 95 cm
b) 1.63 m
a) 20 cm
c) 1630 cm
b) 2000 mm
1.2 CONVERTION OF METRIC UNITS
Metric system is based on multiples of ten.
To change from one unit to another, we must multiply or divide by ten.
c) 2 m
1.2 CONVERTION OF METRIC UNITS
Metric system is based on multiples of ten.
To change from one unit to another, we must multiply or divide by
ten
To change LARGER UNITS TO SMALLER UNITS, multiply by 10 for every
place moved to the right.
To change SMALLER UNITS TO LARGER UNITS, divide by 10 for
every place moved to the left.
1. LENGTH
Measurements of 1 dimension
To change LARGER UNITS TO SMALLER UNITS, multiply by 10 for every place
moved to the right.
To change SMALLER UNITS TO LARGER UNITS, divide by 10 for every place moved
to the left.
For example change 50 cm to mm.
50 x 10 (1 place to the right) = 500 mm.
cm (larger) mm (smaller)
700dm to m
70 : 100 (2 places to the left) = 0,7 m
dm (smaller)
m (larger)
Exercise:
Complete: a) 1,2 km =_______m
d) 0,5 dm= _______mm
b) 4 m = _____ cm
e) 2 dam = _____km
c) 25 cm = ______m
f) 1550 mm = _____________hm
1.3 PRACTICES of Length
EXERCISE 1 Drawing dictation (materials needed: a ruler)
To the whole class: ask them to draw lines or figures using rulers. Give them five minutes
Example: A)draw a line of 4,5 cm
D) Draw a tree of 40 mm tall
B) Draw a line of 30mm
C) draw a line of 1,5dm
E) Draw an envelope of 5cm of length and 35mm width
Correct in class checking in groups or at the blackboard.
GAME: Students propose similar dictations in groups or to the whole class as a BINGO.
EXERCISES 2 How tall are you? (materials needed: ruler and tape measure)
Divide the class in groups of 4 or 5 students. Who is the tallest of the group? Who is the
smallest of the group? Use a tape measure and show your answers with a drawing.
In my group the tallest is David. He is 1,56 m tall and the shortest is Lucas. He is
1,35 m tall.
Lucas
David
1,56
Express
m
the
tall
measurements in dm, cm, mm.
1,35 m tall
2. MASS
Investigate how heavy an elephant, an ant, a cow…are.,
What units of mass do we use to express their weight? Tons, kilograms, grams
Mass is a measurement of how heavy something is.
The basic unit of mass in the metric system is gram (g).
Units of mass. Multiples and submultiples are:
Kilogram
Hectogram Decagram
1kg=
1000g
1 hg =
100g
gram
1 dag =
10g
decigram
centigram
milligram
1dg = 0,1g
1cg =
0,01g
1mg=
0,001g
Multiples
Submultiples
To measure mass we can use scales (metric balance).
Examples: - A dictionary has a mass of about 1 kg
- A paper clip has a mass of about 1 g
- A grain of salt has a mass of about 1 mg
- The mass of heavy things is expressed in tons (t)
1 ton = 1000 kg
We use also “quintal” (q) for 100 kg. Search on Internet other units of mass in Spain: like
arroba, fanega…
Exercise: Choose one of these units (g, mg, kg, t) to express the mass of:
a) a bag of potatoes
b) a box of cereal c) a feather d) a hamster e) a lorry
To convert one unit to another proceed as in length: count the number of places and
multiply or divide by powers of 10.
Example: 150g to kg there are 3 places to the left, so divide by 1000 150:1000= 0,150kg
150g to mg there are 3 places to the right, so multiply by 1000
150 x 1000 = 150 000 mg
Exercise: compare, write <, =, or > a) 754 kg _____754 000g
c) 280 000 mg____ 28 g
d) 1845hg ____18,45 kg
b)876 hg____8.96 kg
e) 0,0001 g ____10 mg
Propose similar exercises to your teams and correct them together.
Complete this converting unit table:
0,854 kg
kg
hg
dag
g
dg
cg
mg
0,854
8,54
85,4
854
8540
85400
854000
324,54 g
910 dag
2t
2.1 PRACTICES of Mass
EXERCISE 1 Body mass index (materials needed: bathroom scales and tape measure)
To calculate the body mass index we use this formula:
weight
height 2
Use a bathroom scale to measure your weight and a tape measure to measure your height,
then divide the weight by the square of your height.
47
47
=
= 20,346
1,52 2 2,31
Paula weight 47 kg
Height 1,52 m
20,35
to know more visit:
http://www.nhlbisupport.com/bmi/bmi-m.htm
EXERCISE 2 How heavy is…? (materials needed: scales)
Divide the class in groups and ask students to measure the weight of school materials and
their lunch. They have to choose 5 items and then change them to different mass units.
Item
Math book
School bag
Sandwich
Biscuits
g
kg
mg
hg
2. MASS
Investigate how heavy an elephant, an ant, a cow…are.
What units of mass do we use to express their weight? Tons, kilograms, grams
Mass is a measurement of how heavy something is.
The basic unit of mass in the metric system is gram (g).
Metric units of mass are related to each other in the same way that
place-value positions within the decimal system of numeration are
related.
The mass of heavy things is expressed in tons (t)
1 ton = 1000 kg.
We use also “quintal” (q) for 100 kg.
3. CAPACITY
It’s the amount of liquid a container can hold. (Also volume) We should drink 2
litres of water per day. If we have glasses of ml. How many glasses will we
fill?
2 litres = 2000 ml 2000 : 200 = 10 glasses of water.
To measure how much a container can hold we use the units of capacity
Example: The water in a swimming pool is measured in kilolitres.
A tall thermos holds about 1 L. 20 drops of water equals 1 mL To
measure capacity we can use a measuring cylinder or a beaker.
Practise to measure little things as jewellery or very light things like a feather, medicines,
spices… with a special scale: precision balance or scale.
3. CAPACITY
It’s the amount of liquid a container can hold. (Also volume)
We should drink 2 litres of water per day. If we have glasses of
ml. How many glasses will we fill?
2 litres = 2000 ml
200
2000 : 200 = 10 glasses of water.
To measure how much a container can hold we use the units of capacity:
Example: a carton holds 1 litre of milk: it has the capacity of 1 litre.
The litre is the basic unit of capacity in the metric system. It represents
what a cube of 1 dm of side can hold. So 1 dm3= 1 L
Units of capacity. Multiples and submultiples are:
Kilolitre
Hectolitre
Decalitre
1kL=
1000L
1 hL =
100L
1 daL=
10L
Litre
L
Multiples
decilitre
1dL
0,1L
centilitre
= 1cL =
0,01L
millilitre
1mL=
0,001L
Submultiples
Example: The water in a swimming pool is measured in kilolitres.
A tall thermos holds about 1 L.
20 drops of water equals 1 mL
To measure capacity we can use a measuring cylinder or a beaker.
Give more examples of measurements of capacity.
To convert one unit to another: count the number of places and multiply or divide by
powers of 10.
Covert 2 L to KL = 3 places to the left
Divide by 10 raised to 3
2 : 103 = 0,002 kL
Example 5L = 0,005kL = 0,05 hL = 0,5 daL = 50 dL = 500cL = 5000 mL
Exercise:
1. Convert these units: a) 8,2 kL to L
b) 65,4 mL to L
c) 45 daL to mL
2. Express in litres the following measurements:
a) 3 daL, 6 L and 2 dL
b) 7hL, 2 L and 8 mL
c) 3 kL, 20cL and 300 mL
2.1 PRACTICES of Capacity
EXERCISE 1 Finding the volume of irregularly shaped object. Displacement
method. Investigate how Archimedes discovered this method.
1. Pour water into a beaker. Read the water level. 50 mL
2. Drop a stone into the water. Read the new water level. 65mL
3. Subtract 65 – 50 = 15 mL is the volume of the stone.
EXERCISE 2 Liquid products
Bring to the class labels of different liquid products, for example: empty bottles of
medicines, cleaning products, cans of drinks, cartons of milk or juice, yogurt…
How much liquid can they hold? Express it in different units and show on a table.
Item
L
kL
mL
cL
A bottle of bleach
A bottle of shampoo
A can of coke
A brick of juice
EXERCISE 3 Problems in daily life.
1. I have 36 containers of 15 L of olive oil to sell to the market.
rest in bottles of 1 L to s. How many bottles of 1 L will I have?
2
are full and I pour the
3
2. How many glasses of 330 mL do we need to fill a bottle of 2 L?
3. From one orange I get 5 cL of juice. How many oranges will I need for
one litre of juice?
4. For a party I bought 5 cans of 33 cL of coke, 2 bottles of 2 litres of lemonade and 10
cartons of 125 mL of juice. How much liquid did I buy?
1
1
2
of orange juice,
of lemon juice, of pineapple
8
10
3
juice and the rest of water. If I want to prepare 1 litre, how much of each juice will I use?
5. The recipe for a cocktail says:
Solve them in groups and invent similar ones to your class group.
-To convert between units of area,
multiply or divide by squares of
powers of ten
When converting from bigger to smaller units,
multiply.
When converting from smaller to larger
units, divide.
5. AREA Measurements of 2 dimensions 10 X 10 = 100
In a flat shape is the space inside the lines.
What’s the measure of your blackboard?
To find it out we measure the area (green part) using surface units. We have to measure 2
dimensions:
Length and width and multiply them: 2m long x 1,20 m wide = 2,40 m2
The basic unit of surface in the metric system is the squared metre, and it is the area
of a square of 1 m of side: 1 m2
Units of surface: Multiples and submultiples are:
Square
Square
Square
Kilometre Hectometre Decametre
1km2=
1 hm2 =
1000 000
10 000
m2
Square
metre
1 dam2=
100 m2
Square
Square
Square
decimetre
centimetre
millimetre
1dm2 = 1cm2 =
0,001m2
0,000 01 m2
m2
1mm2=
0,0000
001m2
m2
Multiples
Submultiples
To convert units of surface we multiply or divide by 100 for each place it depends on if we
move to the right or to the left. For example to change 2 hm2 to m2, as we change from a
greater to a lowest we multiply by 100 as many times as places. Since we move 2 places to
the right from hm2 to m2 we should multiply by 10 000 = 2 x 10 000 = 20 000 m2.
Example: A table is 120 cm long and 70 cm wide.
Its area is 120 x 70 = 8400 cm2 = 8400 : 100 00= 0,84 m2
Área and hectárea
en español. ¿y
fanega?
Other units for large surfaces: area 1(a) = 100 m2
And hectare 1(ha) = 1hm2 =100 areas = 10 000m2
Exercises:
1. What unit of surface will you use to measure: cm2, mm2, km2, m2, dm2?
a) Your room floor
2. Complete the table:
b) a paper area
Km2
hm2
c) The surface of a country
dam2
m2
7
5
0,0025
dm2
cm2
70000
mm2
To convert between units of volume,
multiply or divide by cube of powers of ten
When converting from bigger to smaller units, multiply. . .
When converting from smaller to larger units, divide.
2.1 PRACTICES of surface
EXERCISE 1 Area of objects (materials needed: ruler and tape measure)
Complete this table. In groups they have to measure length and width of these objects to
find the area. Remember to use the same units to operate. Compare the results with other
groups (give them 10 minutes maximum).
AREA in m2
Length
Width
AREA
2m
70cm
200 x 70 = 14000:100 00=
14000 cm2
1,4 m2
OBJECT
Door
Window
Ruler
Math book
Classroom
Paper sheet
EXERCISE 2
Space at home.
Measure at home the area of your bedroom and other rooms from your house. Then
prepare problems to show to your group or class.
For example: My bedroom is 230 cm wide and 4,5 m long. If I want to covert the floor
with a blue carpet, how many m2 will I need?
EXERCISE 3 Surfaces of places
Find out the surface of different countries or towns; express it in m2 and compare.
Example: Spain has a surface of 506,990 km2, is it bigger than France?
If we have 30% of forest surface, How many hectares are there?
30% of 506,990 = 152,097 km2 = 152,097 x 100 = 15209,7 ha
4. VOLUME Measurements of 3 dimensions 10 X 10 X 10 = 1000
Generally,volume is measurement of solids.In a 3 D shape or
solid shape, the volume is how much space something occupies.
We use a cube as a measurement unit.
The basic unit of volume in the metric system is the cube metre, and it is the space
that takes a cube of 1 m of side: 1 m3
Units of volume. Multiples and submultiples are:
Cubic
Cubic
Kilometre Hectometre
1km3=
1 hm3 =
1000 000 1000 000
000 m3
m3
Cubic
Decametre
3
1 dam =
1000 m3
Cubic
Cubic Cubic
metre decimetre centimetre
m3
3
3
1dm
= 1cm = 0,000
3
0,001m
001 m3
Multiples
Cubic
millimetre
1mm3=
0,0000
000
001m3
Submultiples
To convert units of volume we multiply or divide by 1000 for each place it depends on if
we move to the right or to the left. For example to change 2 hm3 to m3, as we change from
a greater to a lowest we multiply by 1000 as many times as places. Since we move 2 places
to the right from hm3 to m3 we should multiply by 1000 000 = 2 x 1000 000 = 2000 000
m3.
Example: An object has a volume of 245 cm3, how many dm3 ? and mm3?
245 : 1000 = 0,245 dm3
245 x 1000 = 245 000 mm3
Exercise:
Express in m3 the following volume: a) 0,4 hm3
b) 0,0032 dm3 c) 24 dm3
4.1 RELATING METRIC UNITS
Metric units of volume, capacity and mass are related to one another in this way:
A cube of 1 dm of side holds 1 litre of water and has a mass of 1 kg (Only for water o
similar liquids density, for example not for honey)
1 dm3 = 1L of water = 1 kg
Exercise: How many litres are in 3 m3?
1st we convert m3 to dm3
3 x 1000 = 3000 dm3 = 3000 L
Solved problem: A child wants to fill a 500 cm3 bucket. How many litres of water does he
need to fill it?
500cm3 dm3
litre or 500 mL
1 place to the left, so divide by 1000 = 0,500 dm3 = 0,5 L what is half a
2. On this signpost, the distance to Madrid is given as 9 miles. Show the distance in km.
Madrid 9
Airport
Invent similar problems to your class and check the results.
http://www.math-drills.com/measurement.shtml Metric- imperial conversion exercises.
III. MEASURING TEMPERATURE
We have three scales to measure temperature: CELSIUS, FAHRENHEIT AND
KELVIN.
Celsius is used In Spain and many countries in Europe and Fahrenheit is used in U.S.A.
http://www.bbc.co.uk/skillswise
They settle in different degrees the freezing point of water and the boiling point of water.
CELSIUS
O
C
FAHRENHEIT
Freezing point of
water
0 oC
32 o F
Boiling point of
water
100 0 C
212 o F
Normal Body
temperature
37 o C
98,6 o F
O
F
5
5
= Example 78oF
(78 – 32) = 25,5 o C
9
9
Twenty-five point five degrees Celsius or Centigrade
To convert o F to o C
(oF – 32)
9
+ 32 = Example 32 o C
5
Eighty-nine point 6 degrees Fahrenheit
To convert o C to o F
o
C
32 x
9
+32 = 89,6 o F
5
Why the fraction 5/9 and 9/5? The difference between the freezing point and the boiling
point in Fahrenheit is 212 – 32 = 180, in Celsius is 100 – 0 = 100. The proportion is
o
C to o F =
180 9
=
+ 32 if we reduce it.
100 5
And o F to o C =
100 5
= – 32
180 9
Converting practices and reading temperatures: http://www.mathdrills.com/measurement.shtml
Solve these problems and convert the degrees to Celsius. Invent similar ones for the class.
Example: At 11 a.m. the temperature outside is 35 o F, and at 11 p.m. is – 13 o F. By how
many degrees has the temperature fallen?
35 – (–13) = 35 + 13 = 48 o F
(48 –32) 5/9 = 8,8 o C
1. The pool water temperature at 9 a.m. was 62 oF, but by 6 p.m. the temperature was 70
o
F, How much has the temperature risen?
2. When Susan was sick, the temperature was 38,8 o C. After she recovered, her
temperature was 36,5 o C. How much has her temperature dropped?
Exercise 2: Write “R” if the statement is reasonable and “U” if it is unreasonable:
a) Your body temperature when you are well is about 37 oC ______
b) Inside a freezer it is 10 oC _______
c) The skating lake is frozen when the temperature is – 5 o C _____
d) You need a coat in 25 o C ______
e) When you boil potatoes the water is about 70o C ______
Exercise 3: Choose a reasonable temperature for:
1) The temperature of your classroom in a warm day: a) 68 o F
2) The temperature of a dish of ice cream: a) 31 o F
b) 80 o F
b) 0 o F
c) 45 o F
c) –10 o F
PRACTICE 1 Weather report
Bring newspapers or documents where the students can read the weather report and
temperatures.
In groups they choose a country or area and they should convert the temperature in degrees
Fahrenheit. Then they will write and read aloud a report about the temperatures from
yesterday to tomorrow in
Celsius degrees and
Yesterday the temperature in
In degrees Fahrenheit:
Fahrenheit
Oxford was 7oC, Today the
Yesterday
was 44,6oF,
degree:
o
o
temperature will be 10 C and
Tomorrow it will be 8 oC
today 50 F and tomorrow
it will be 46,4 oF
To convert units of surface we multiply or divide by square of power of ten .
For example to change 2 hm 2 to m2 , as we change from a greater to a lowest
we multiply by 100 as many times as places. Since we move 2 places to the right
from hm 2 to m 2 we should multiply by 10 000 = 2 x 10 000 = 20 000 m2 .
Example: A table is 120 cm long and 70 cm wide.