Geographical Area
Measurements
Objectives To discuss how geographical areas are measured;
and
to provide practice using division to compare two quantities
a
with like units.
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Teaching the Lesson
Key Concepts and Skills
• Use division to compare two quantities
with like units. [Operations and Computation Goal 4]
• Use “times as many” language to compare
area measurements. [Operations and Computation Goal 4]
• Estimate and compare area measurements. [Measurement and Reference Frames Goal 2]
Key Activities
Students examine how geographical areas
are measured and the difficulties involved
in making accurate measurements. They
compare the areas of different countries by
guessing and then using division to calculate
the ratio of areas.
Materials
Math Journal 2, p. 245
Student Reference Book, pp. 286, 287,
and 295
Study Link 8 7
world map or globe calculator slate
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Grab Bag
Student Reference Book, p. 249
Grab Bag Cards (Math Masters,
pp. 483 and 484)
Math Masters, p. 485
3 six-sided dice
Students practice calculating the
probability of an event.
Ongoing Assessment:
Recognizing Student Achievement
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Comparing Areas
Math Masters, p. 270
scissors
Students explore area comparison using
a concrete model.
ENRICHMENT
Calculating Gravitational Pull
Math Boxes 8 8
Math Masters, p. 271
calculator
Students use patterns in a table of weights
to determine the gravitational pull of each
planet relative to Earth.
Math Journal 2, p. 246
Students practice and maintain skills
through Math Box problems.
Exploring Similar Figures
Use Math Masters, page 485. [Data and Chance Goal 4]
Study Link 8 8
Math Masters, p. 269
Students practice and maintain skills
through Study Link activities.
ENRICHMENT
Math Masters, pp. 272 and 273
centimeter ruler
Students explore the relationships between
the dimensions and areas of similar figures.
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 231, 232
Lesson 8 8
699
Mathematical Practices
SMP2, SMP3, SMP4, SMP5, SMP6, SMP7
Content Standards
Getting Started
4.OA.2, 4.OA.3, 4.NBT.3, 4.MD.2
Mental Math and Reflexes
Write numbers on the board for students to round to various places. Suggestions:
Round 1,005,518 to the nearest
10,000 1,010,000
100,000 1,000,000
1,000,000 1,000,000
Round 440,762 to the nearest
100 440,800
10,000 440,000
100,000 400,000
Round 293,571,551 to the nearest
100,000 293,600,000
1,000,000 294,000,000
100,000,000 300,000,000
Math Message
Study Link 8 7 Follow-Up
Read page 295 of the Student Reference Book.
Be prepared to give several reasons why it is hard
to measure the areas of countries, oceans, and deserts.
Have small groups compare answers and discuss
how they solved Problems 5 and 6. Sample
answer: Multiply the area by 2. Then divide that
number by the height to find the length of the base.
1 Teaching the Lesson
Math Message Follow-up
WHOLE-CLASS
DISCUSSION
(Student Reference Book, p. 295)
Listed below are some ideas that should emerge from a discussion
of Student Reference Book, page 295.
It is often difficult to make accurate area measurements of
land forms.
People do not always agree on where the borders of a country
are located, or even on the definition of a land form.
Student Page
World Tour
Geographical Area Measurements
The heights of mountains and the depths of oceans are
obtained directly. We find heights and depths by measuring the
Earth itself.
Students may be surprised to learn that places where the water is
always frozen, called tundras, are actually considered deserts.
The areas of countries and the areas of oceans are found
indirectly. We measure very accurate maps or satellite pictures.
The countries and oceans themselves are not measured.
Countries, oceans, and deserts have irregular boundaries. One
way that scientists measure areas is to count grid squares. They
place a transparent grid of squares on a map. Then they count
the squares and parts of squares that cover the region being
measured. The squares are drawn to the same scale as the map.
There are several reasons that it is hard to measure the
following regions accurately:
Area of a country. Sometimes people disagree about the
exact boundary of a country. So the area may depend on which
boundary is being used.
Tell students that in this lesson they will use division to compare
areas of countries to determine relative size.
The shoreline of a body
of water may shift greatly
during different seasons
of the year and over
the years.
Area of a lake, sea, or ocean. Some bodies of water have
shorelines that shift greatly depending on the level of the
water. So it is very hard to measure accurately the area that
is covered by water.
Area of a desert. Measuring desert areas is very hard. Desert
boundaries may change because the climate changes. When
land is cultivated, a desert boundary shifts. Also, scientists do
not agree on what a desert actually is. Some define a desert as
land that cannot be used for raising crops. Others define it as
land that cannot be used for either crops or grazing. There are
deserts that are hot and dry only part of the year. Some deserts
are dry all year because it is very hot. Other deserts are dry all
year because it is very cold and the water is always frozen.
Very cold deserts are known as tundras.
Desert boundaries
often change because of
climate changes.
700
Unit 8 Perimeter and Area
Links to the Future
Students used division to solve equal-sharing and equal-grouping problems in
Unit 6. This lesson introduces a third use of division: to compare two quantities
that are measured with the same unit. In Unit 9, students will have additional
experiences with ratio comparisons.
The world’s oceans are not separated from one another by
shorelines. Sometimes people disagree on the boundaries between
the oceans. This makes it difficult to measure the areas of oceans.
Student Reference Book, p. 295
Boundaries may change due to cultivation of the land, political
events, or changes in climate.
Time
290,000
11.4
3,300,000 ÷
mi2
290,000
292,300 mi2
Chile
=
mi2
70,000
68,000 mi2
Uruguay
=
47.1
3,300,000 ÷
70,000
500,000
6.6
3,300,000 ÷
mi2
500,000
496,200 mi2
Peru
=
160,000
20.6
3,300,000 ÷
mi2
160,000
157,000 mi2
Paraguay
=
3.1
3,300,000 ÷
=
2
1,070,000 mi
1,068,300 mi2
110,000
30
3,300,000 ÷
=
mi2
110,000
109,500 mi2
(3)
Area
(rounded to the
nearest 10,000)
(2)
Area
(1)
Argentina
Have students find Ecuador on the map. Use the following routine
to compare the areas of Ecuador and Brazil. Students fill in the
first line of the table on journal page 245 as you work through
the steps.
Guess the number
of times it would fit
in the area of Brazil.
Tell students that they will be comparing the areas of other
countries in South America to Brazil’s area. Since Brazil and the
United States have nearly the same area, these comparisons will
be nearly the same as if they had compared the areas of other
South American countries to the area of the United States.
Country
Social Studies Link Ask students to turn to the map of
South America on pages 286 and 287 of the Student
Reference Book. The country in South America with the largest
area is Brazil. Use the classroom world map or a globe to compare
Brazil and the United States and mention that they have nearly
the same area. (The United States is about 10% larger.)
(4)
PROBLEM
PRO
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
LV
VIN
IIN
NG
1,070,000
Comparing Country Areas
Answers
vary.
Ecuador
88
Divide the rounded areas.
(Brazil area ÷ country area)
LESSON
Fill in the table below. This will help you to compare the areas of other countries in South America
to Brazil’s area. Round quotients in Part 4 to the nearest tenth.
(Math Journal 2, p. 245; Student Reference Book,
pp. 286 and 287)
WHOLE-CLASS
ACTIVITY
Brazil is the largest country in South America. Brazil’s area is about 3,300,000 square miles. The area of
the United States is about 3,500,000 square miles. So Brazil is nearly the same size as the United States.
Comparing Country Areas
Student Page
Date
Math Journal 2, p. 245
219-247_EMCS_S_MJ2_G4_U08_576426.indd 245
2/1/11 1:47 PM
Guess how many times larger Brazil is than Ecuador. Ask
students to imagine that they have many paper cutouts that
are the size and shape of Ecuador. About how many cutouts
would it take to cover Brazil? Said another way, how many
Ecuadors would fill up Brazil? Expect answers that range
from 20 to 50.
Round the areas of Brazil and Ecuador to the nearest
10,000 square miles. The area of Ecuador is given in column (2)
on the journal page as 109,500 square miles. The rounded area
is 110,000 square miles. Brazil’s area is reported (at the top of
the journal page) as 3,300,000 square miles, which is already
rounded to the nearest 10,000 square miles.
Estimate how many times larger Brazil is than Ecuador. Point
out that students need to figure out how many 110,000s there
are in 3,300,000. Write 3,300,000 / 110,000 = ? on the board,
and have students use their calculators to divide. Brazil is
about 30 times the size of Ecuador. About 30 cutouts or copies
of Ecuador would fit inside the boundary of Brazil.
Adjusting the Activity
Have students explore the use of the
constant feature on their calculators to do the
repeated divisions in which the dividend is
3,300,000.
AUDITORY
KINESTHETIC
TACTILE
VISUAL
Have partnerships complete journal page 245. Make sure they
understand that the different country areas are always compared
with Brazil’s area. You might want to point out to students that
when they complete journal page 245, they are using division to
solve real-world multiplicative comparison problems.
In most cases, the division of Brazil’s area by the area of another
country will lead to a decimal answer. For example, Brazil’s area
divided by Peru’s is 3,300,000 mi2 / 500,000 mi2 = 6.6.
Lesson 8 8
701
Student Page
Date
Time
LESSON
Make scale drawings of each rectangle
described below.
1.
2 Ongoing Learning & Practice
Math Boxes
88
a.
Scale: 1 cm represents 1.5 meters.
a.
Length of rectangle: 6 meters
Width of rectangle: 3 meters
b.
Length of rectangle: 10.5 meters
Width of rectangle: 4.5 meters
Playing Grab Bag
(Student Reference Book, p. 249; Math Masters, pp. 483–485)
b.
145
What is the area of the parallelogram?
2.
3.
Algebraic Thinking Students play Grab Bag to practice calculating
probabilities of events. See Lesson 7-6 for additional information.
A jar contains
27 blue blocks,
8"
PARTNER
ACTIVITY
18 red blocks,
1"
12 orange blocks, and
1∗8=8
Number model:
8
Area =
43 green blocks.
You put your hand in the jar and without
looking pull out a block. About what
fraction of the time would you expect to
get a red block?
2
in
18
_
100 ,
135
Add or subtract.
4.
1 +_
11 =
_
a.
12
1
b. _
6
12
+ _2 =
5.
12
_
12 , or 1
_5
3
c.
_2 , or _1
8
4
3
_
d.
16
or
9
_
50
Ongoing Assessment:
Recognizing Student Achievement
45
Multiply. Use a paper-and-pencil algorithm.
91 ∗ 48 =
Use Math Masters, page 485 to assess students’ ability to calculate the
probability of an event. Students are making adequate progress if they are
able to calculate the total number of items in the bag and express the probability
of an event as a fraction. Some students may use a strategy when replacing
x and y to earn the most possible points for each turn.
4,368
6
= _7 - _5
8
8
5 -_
1
=_
16
8
55 57
Math Masters
Page 485
[Data and Chance Goal 4]
18 19
Math Journal 2, p. 246
219-247_EMCS_S_MJ2_G4_U08_576426.indd 246
2/1/11 1:47 PM
Math Boxes 8 8
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 246)
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 8-6. The skill in Problem 5
previews Unit 9 content.
Writing/Reasoning Have students write a response to the
following: Explain the strategy you used to solve Problem 4d.
2
2
Sample answer: _18 is equivalent to _
. I renamed _18 as _
so that
16
16
I would have two fractions with the same denominator.
5
3
2
_
-_
=_
16
16
16
Study Link Master
Name
Date
STUDY LINK
Study Link 8 8
Time
Turtle Weights
88
Turtle
Weight (pounds)
Pacific leatherback
1,552
Atlantic leatherback
1,018
Green sea
783
Loggerhead
568
Alligator snapping
220
Flatback sea
171
Hawksbill sea
138
Kemps Ridley
133
Olive Ridley
110
Common snapping
(Math Masters, p. 269)
Home Connection Students compare the weight of the
10 heaviest turtles. They use data in a table to estimate
answers to given questions.
85
Source: The Top 10 of Everything 2004
Olive Ridley turtle.
1.
The Atlantic leatherback is about 10 times heavier than the
2.
The loggerhead is about
3.
Which turtle weighs about
3 times as much as the loggerhead?
4.
The flatback sea turtle and the alligator snapping
turtle together weigh about half as much as the
Green sea
5.
About how many common snapping turtles would
equal the weight of two alligator snapping turtles?
5
6.
7
times the weight of the common snapping turtle.
Pacific leatherback
turtle.
2
The Atlantic leatherback is about _ the weight of the Pacific leatherback.
3
Practice
Name the factors.
7.
50
9.
90
1, 2, 5, 10, 25, 50
1, 3, 7, 9, 21, 63
8. 63
1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Math Masters, p. 269
247-277_EMCS_B_MM_G4_U08_576965.indd 269
702
INDEPENDENT
ACTIVITY
Unit 8 Perimeter and Area
2/1/11 2:17 PM
Teaching Master
Name
3 Differentiation Options
Date
LESSON
Time
Weight on Different Planets
88
Mercury has about _3 the gravitational pull on your body mass as does Earth — about 0.37
1
to be more precise. You would weigh about _3 as much on Mercury as you do on Earth.
1
The table below shows how much Rich, his brother Jean-Claude, and his sister Gayle
would weigh on each planet.
SMALL-GROUP
ACTIVITY
READINESS
Comparing Areas
Use your calculator to find each planet’s gravitational pull relative to Earth’s.
1.
Weight in Pounds
15–30 Min
Planet
(Math Masters, p. 270)
Gravitational
Pull Relative
to Earth’s
Earth
Mercury
Venus
To explore area comparisons using a concrete model, have
students cut out the shapes on Math Masters, page 270 and
describe relationships among them.
Mars
Jupiter
Saturn
Uranus
Neptune
Calculating Gravitational Pull
1
86
0.37
0.90
0.37
2.35
0.91
0.88
1.12
Gayle
75
50
31.82
27.75
18.5
77.4
67.5
45
31.82
27.75
18.5
202.1
176.25
117.5
78.26
68.25
45.5
75.68
66
44
96.32
84
56
Me
Source: Nasa Kids
INDEPENDENT
ACTIVITY
ENRICHMENT
JeanClaude
Rich
Explain the strategy you used to determine the gravitational pulls.
2.
Sample answer: I divided Rich’s weight on
each planet by his weight on Earth.
5–15 Min
(Math Masters, p. 271)
Try This
Use the information in the table to calculate your own weight on each planet
and record it in the “Me” column in the table above.
3.
Answers vary.
Science Link To apply students’ understanding of
comparison strategies, have them use patterns in a table
of weights to determine the gravitational pull of each planet
relative to Earth.
Math Masters, p. 271
INDEPENDENT
ACTIVITY
ENRICHMENT
Exploring Similar Figures
5–15 Min
(Math Masters, pp. 272 and 273)
To apply students’ understanding of area comparisons, have them
explore perimeters and areas of similar figures.
Teaching Master
Name
LESSON
8 8
Teaching Master
Date
Time
Name
Similar Figures
Date
LESSON
88
Imagine that you used a copying machine to enlarge the original figures below and on
Math Masters, page 273 to get similar figures. Find the perimeter of each original shape
and of its enlargement.
Similar Figures
Time
continued
1 cm
Original
3.
Enlargement
1 cm
Original
1.
Enlargement
10
Perimeter =
c.
How many small rectangles can fit inside the large rectangle?
cm
b.
Perimeter =
20
a.
cm
4
Use a centimeter ruler to measure the longest side of each triangle.
Draw the small rectangles inside the large rectangle.
2.
Original
Enlargement
12
Perimeter of original =
b.
Perimeter of enlargement =
c.
How many small triangles can fit inside the large triangle?
d.
Draw the small triangles inside the large triangle.
cm
24
cm
4
p
py g
a.
g
g
4.
Complete the statements.
p
When you enlarge the sides of a shape to twice their original size, the
2
perimeter of the enlargement is
times as large as the perimeter
of the original shape.
b.
When you enlarge the sides of a shape to twice their original size, the area
4
of the enlargement is
times as large as the area of the original
shape.
py g
a.
12
Perimeter =
Area =
5
cm2
cm
24
20 cm
Perimeter =
Area =
cm
2
Math Masters, p. 272
247-277_EMCS_B_MM_G4_U08_576965.indd 272
Math Masters, p. 273
2/1/11 2:17 PM
247-277_EMCS_B_MM_G4_U08_576965.indd 273
2/1/11 2:17 PM
Lesson 8 8
703