Estimating Distances
with a Map Scale
Objective To introduce the use of a map scale to
estimate distances.
e
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
Key Concepts and Skills
Areas of Rectangles
• Solve problems involving fractional parts
of regions on a map scale. Math Journal 1, pp. 96A and 96B
Student Reference Book, pp. 154,
155, 156A, and 156B
Children practice multiplying side
lengths to find the areas of rectangles.
[Number and Numeration Goal 2]
• Fill in missing numbers on number lines. [Number and Numeration Goal 6]
• Use multiplication strategies to solve
map-scale problems. [Operations and Computation Goal 4]
• Use map scales to estimate the most
direct distance between two places. [Measurement and Reference Frames Goal 1]
Key Activities
Children fill in missing numbers on a number
line. Children use map scales to estimate the
most direct distance between two places.
Ongoing Assessment:
Recognizing Student Achievement
Use the Math Message. [Number and Numeration Goal 6]
Key Vocabulary
map scale as the crow flies scale factor
Materials
1 2
4 3
Playing Division Arrays
Student Reference Book, p. 282
per group: 1 each of number
cards 6–18 (from the Everything
Math Deck, if available), 18 counters,
1 six-sided die
Children practice basic division facts
as well as recognition of even and
odd numbers.
Curriculum
Focal Points
Differentiation Options
READINESS
Exploring Scale
per partnership: toy animals, large paper,
crayons, research materials
Children compare the sizes of toy animals
to the actual sizes of real animals.
ENRICHMENT
Making Scale Drawings
Math Masters, p. 112
ruler
Children draw classroom objects to scale.
Math Boxes 4 9
Math Journal 1, p. 97
Children practice and maintain skills
through Math Box problems.
Home Link 4 9
Math Masters, pp. 110 and 111
Children practice and maintain skills
through Home Link activities.
Math Journal 1, pp. 95 and 96
Home Link 48
Math Masters, p. 109 (optional)
map display straightedge blank sheet of
paper or tinted transparency half-sheet
of paper
Advance Preparation
Display maps (see Planning Ahead, page 287). Include the classroom U.S. High/Low Temperatures map.
Cut enough copies of Math Masters, page 109 so that each child has one paper strip (optional). Copy the
Home Link Masters (Math Masters, pages 110 and 111) on two separate pages so that children have the
problems and the maps side-by-side.
Teacher’s Reference Manual, Grades 1–3 p. 184
288
Unit 4
Multiplication and Division
Interactive
Teacher’s
Lesson Guide
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP7
Content Standards
Getting Started
3.MD.6, 3.MD.7b, 3.MD.7d
Mental Math
and Reflexes
Math Message
Home Link 4 8
Follow-Up
Copy the number lines. Fill in
the missing numbers on a half-sheet
of paper.
Pose multiplication facts. Suggestions:
1 × 3 3 2 × 3 6 5 × 3 15 10 × 3 30
6 × 3 18 7 × 3 21 8 × 3 24 9 × 3 27
30 × 1 30 30 × 2 60 30 × 3 90
30 × 4 120
0
400
0
1
Before turning in their
work, have partners share their answers.
Have them compare their arrays and
answers for Problems 3 and 4.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Children share their results and their thinking. Then ask them to
extend the first number line in both directions and the second
number line to the right.
Adjusting the Activity
−100
100
200
300
400
500
Number line extended by 100s
1
−–
2
Have children extend the second number line to the left as well as to
0
1
− –
4
0
1
–
4
1
–
2
3
–
4
11–4
1
11–2
Number line extended by _14 s
the right.
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
Ongoing Assessment:
Recognizing Student Achievement
V I S U A L
Math
Message
Use the Math Message to assess children’s progress toward using number lines
to order whole numbers and fractions. Children are making adequate progress if
they are able to fill in the first number line. Some children may be able to fill in
the second number line.
[Number and Numeration Goal 6]
Introducing Map Scales
WHOLE-CLASS
ACTIVITY
ELL
Point out the maps on display. Children might notice that the
maps do not cover the same areas. For example, a map of a city
may be on the same size piece of paper as a map of the United
States. Discuss how that is possible with map scales. Explain
that a map scale is a tool that helps to estimate real distances
between places shown on a map by relating distances on the map
to distances in the real world.
Links to the Future
The activities in this lesson are an early
exposure to map scales. Using a map
scale to estimate distances is a Grade 4
Goal. Using scales to find distances is a
Grade 5 Goal.
Lesson 4 9
289
Teaching Master
Name
LESSON
49
䉬
Date
Time
One-Inch Segments
Adjusting the Activity
If possible, show children two maps of the same area that have
different scale factors. Another option is to show some different-size school
pictures (2 by 3, 4 by 6, and 8 by 10) of the same person.
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
V I S U A L
If the scale is not given on a map, make an approximate scale.
The distance from coast to coast in the United States, for example,
from New York to Los Angeles, is about 2,500 miles as the crow
flies. To support English language learners, take a moment to
discuss the meaning of the expression, as the crow flies. Measure
that distance on the map to make an approximate scale of miles
per inch, rounded off to an easy number.
Cartographers (mapmakers) scale down large areas of land and
space to represent them on paper. Places that are actually
thousands of miles apart appear only inches apart on a map.
People then estimate real distances using map scales.
Math Masters, p. 109
Different maps use different scale factors. On one map, 1 inch
may represent 10 miles; on another map, 1 inch may represent
100 miles. To support English language learners, write scale
factor on the board along with some examples of scale factors
using available maps.
Adjusting the Activity
Have children imagine that they are drawing a floor plan of their kitchen
on notebook paper. Their drawings will have to be scaled down in order to fit on
the paper.
A U D I T O R Y
Student Page
Date
LESSON
49
K I N E S T H E T I C
Estimating Distances
Time
Estimating Distances
T A C T I L E
V I S U A L
WHOLE-CLASS
ACTIVITY
7
on a U.S. Map
5
6
Civil Rights Memorial
Statue of Liberty
White House
Cowboy Hall of Fame
Space Shuttle Launch Site,
Cape Canaveral
Ask children how they might estimate the distance between the
cities with the highest and lowest temperatures for the current
week. Discuss, try, and compare some of their ideas. Model the
following methods in your discussion.
9
6.
7.
8.
9.
10.
Math Journal 1, p. 95
EM3MJ1_G3_U04_79-101.indd 95
290
1 inch represents 300 miles
0
2
1
300 miles
Map Scale
3
4
Mount St. Helens
Santa Ana Zoo
Yellowstone National Park
Pikes Peak
Sears Tower
1.
2.
3.
4.
5.
Locations to Visit
10
8
Talk about as-the-crow-flies distances—the most direct route
from point to point on a map. Actual road distances are longer
than these direct paths. Point out the scale information for the
classroom map.
9
12/20/10 6:23 PM
Unit 4 Multiplication and Division
Student Page
Date
Time
LESSON
Method 1
A Pretend Trip
49
Lay a piece of paper or a tinted transparency between the two
cities, mark the distance on the paper, and then mark off the
measured piece along the map scale while counting the miles
as indicated on the scale.
Pretend that you want to take a trip to see some of the sights in the
United States. Find out about how far it is between locations.
1.
The distance between them is about
900
That is about
2.
Pikes Peak is number
4
That is about
1
3
3.
3
5
7
.
2
.
9
6
The distance between them is about
N
W
6
N
Map Scale
0
Map Scale
S
10
300 miles
That is about
1,800
.
inches on the map.
miles.
E
W
0
E
S
10
300 miles
inches on the map.
6
2
9
5
6
The Civil Rights Memorial is number
Santa Ana Zoo is number
2
inches on the map.
miles.
8
4
7
8
4
1,500
.
.
8
The distance between them is about
1
.
miles.
The White House is number
5
3
9
3
Yellowstone National Park is number
The Cowboy Hall of Fame is number
Try This
4.
7
The Statue of Liberty is number
5
The Sears Tower is number
.
.
2_12
The distance between them is about
Method 2
750
That is about
Mark off the scale repeatedly on the edge of a strip of paper and
then use that strip of paper to read the distances directly.
5.
Make up one of your own.
inches on the map.
miles.
Sample answers:
Mount St. Helens
Pikes Peak
is number
1
4
.
is number
.
1
2 inches on the map.
The distance between them is about
3_
1
That is about
1,050
miles.
1
3
5
7
Math Journal 1, p. 96
3
8
4
5
2
7
9
8
4
6
N
79-101_EMCS_S_SMJ_G3_U04_576353.indd 96
2
10
300 miles
1/26/11 9:11 AM
9
E
W
Map Scale
0
S
6
N
W
Map Scale
0
300 miles
10
E
S
NOTE You may want to tell children that
on Math Journal 1, page 95, the Sears
Tower has been renamed Willis Tower, but
many people continue to refer to it as the
Sears Tower.
Adjusting the Activity
Have children estimate partial distances between the marked
increments. For example, children estimate about two and a half 100-mile
increments as 250 miles.
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
Estimating Mileage
on a U.S. Map
V I S U A L
Student Page
Date
PARTNER
ACTIVITY
PROBLEM
PR
PRO
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
ING
Areas of Rectangles
49
Fill in the blanks.
1.
(Math Journal 1, p. 95; Math Masters, p. 109)
Point out the scale for the map on the journal page. This scale
shows that approximately 300 miles are represented by 1 inch on
the map. Choose two locations on the map, such as the Santa Ana
Zoo and Mount St. Helens, and measure the distance between
them. You may want to have the class try one or both of the
following methods.
Time
LESSON
7 units
2.
5 units
This is a
4 -by- 7 rectangle.
28 square units
Number model: 4 × 7 = 28
This is a
Area =
Area =
3.
6 units
5 -by- 10 rectangle.
50 square units
Number model: 5 × 10 = 50
4.
7 -by- 6 rectangle.
42
Area =
square units
Number model: 7 × 6 = 42
6 -by- 9 rectangle.
54 square units
Number model: 6 × 9 = 54
This is a
Partners use the edge of a piece of paper to mark the distance
between the points as described in the previous activity.
Method 2
Children use the paper strips from Math Masters, page 109,
marked off in 1-inch segments, to read the distance.
5.
This is a
Area =
6.
2 units
3 units
This is a
Area =
×
5 units
4 units
3 -by- 2 rectangle.
6 square units
Number model:
3
9 units
6 units
7 units
Method 1
10 units
4 units
2
=
4 -by- 5 rectangle.
20
square units
Area =
This is a
Number model:
6
4
×
5
=
20
96A
Math Journal 1, p. 96A
79-101_EMCS_S_SMJ_G3_U04_576353.indd 96A
1/26/11 9:11 AM
Lesson 4 9
291
Student Page
Date
Time
LESSON
Areas of Rectangles
49
Circulate and help as needed. Bring the class together to share
the results and discuss strategies. Come to a consensus for the
estimate. About 900 miles between the Santa Ana Zoo and Mount
St. Helens Have children use a straightedge to draw a line
segment between the two locations and label it with the
estimated mileage.
continued
Follow these steps to find the area of the shape shown below. Show
your work.
1.
Divide the shape below into rectangles.
2.
Use your ruler to measure the sides of each rectangle in centimeters.
Label the sides of the rectangles with the measures.
3.
Find the area of each rectangle.
4.
How can the area of each rectangle help you find the area of the shape?
Sample answer: The rectangles are parts of
the whole shape. If I find the area of each
part and then add the areas together, I can
find the area of the whole shape.
8
4
cm
cm
4
cm
3
cm
4
cm
2
Total area of shape =
4
cm
3
40
Point out that this measurement is an as-the-crow-flies estimate
to the nearest 50 or 100 miles. Explain that such estimates
provide useful information about the relative distances
between locations.
Have the class repeat the procedure with other locations. Use the
map scale to find the distance; draw a line segment between the
locations, and then label it with the distance. Suggestions:
Cowboy Hall of Fame and Civil Rights Memorial
About 600 miles
cm
cm
Statue of Liberty and Santa Ana Zoo About 2,250 miles
square cm
Willis Tower (formerly Sears Tower) and Space Shuttle Launch
Site, Cape Canaveral About 975 miles
Math Journal 1, p. 96B
79-101_EMCS_S_SMJ_G3_U04_576353.indd 96B
2/10/11 2:49 PM
Taking a Pretend Trip
(Math Journal 1, pp. 95 and 96)
NOTE Have children who are unfamiliar
with these locations do some research on
the Internet to determine if they would be
interested in visiting them.
2 Ongoing Learning & Practice
Time
LESSON
49
1.
5×5
1×9
>
>
2.
4×4
6×2
<
2×8
3×7
=
7×3
Complete.
in
out
in
9
15
3
5
21
30
10
Rule
÷3
out
Complete the Fact Triangle and
write the fact family.
7 × 4
4 × 7
28 ÷ 4
7×1
13 56
3.
28 ÷ 7
4.
7
28
28
= 7
•
=
×, ÷
=
7
4
4
55
Draw an array of 28 Xs arranged
in 4 rows.
How many Xs in each row?
7 Xs
4 × 7 = 28 or 28 ÷ 4 = 7
65
median = 70
Playing Division Arrays
64 65
6.
60, 54, 79, 80, 65, 74, 70, 65, 81
Children practice multiplying side lengths of rectangles on journal
page 96A to find the areas. They find the area of a rectilinear
figure on journal page 96B. If necessary, have children read
Student Reference Book, pages 154, 155, 156A, and 156B.
Write a number model for the array.
Answers vary.
Ages of 9 grandfathers:
PARTNER
ACTIVITY
(Math Journal 1, pp. 96A and 96B; Student Reference Book,
pp. 154, 155, 156A, and 156B)
28
=
52
203–204
5.
Areas of Rectangles
Math Boxes
Write <, >, or 5.
PROBLEM
PRO
PR
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
IN
ING
Children work with partners or in small groups to answer
questions about a pretend trip. Circulate among the groups to
offer assistance. Bring the class together to share their strategies
for estimating distances between locations.
Student Page
Date
PARTNER
ACTIVITY
Fill in the empty
frames.
Rule
SMALL-GROUP
ACTIVITY
(Student Reference Book, p. 282)
⫹1,000
mode =
80 81
4,670
5,670
6,670
9,670
8,670
7,670
10,670
200 201
Math Journal 1, p. 97
79-101_EMCS_S_SMJ_G3_U04_576353.indd 97
292
Unit 4 Multiplication and Division
2/4/11 10:24 AM
Children practice basic division facts as well as recognition of even
and odd numbers by playing Division Arrays. If necessary, review
the game directions on page 282 in the Student Reference Book.
Math Boxes 4 9
Home Link Master
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 97)
Name
HOME LINK
49
䉬
Date
Family
Note
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 4-6. The skill in Problem 6
previews Unit 5 content.
Writing/Reasoning Have children write an answer to the
following: Explain how you figured out the number you
wrote in the first empty frame of Problem 6. Sample
answer: I subtracted 1,000 from the number in the second frame.
Time
Using a Map Scale
Your child
Family
Note
is text
just learning how to use a map scale. He or she should use the scale to measure
an
as-the-crow-flies
estimate
eachtomorrow.
problem. This expression refers to the most direct route
Please
return this Home
Link tofor
school
between two points, disregarding road distance. Actual road distances are longer
than these direct paths.
000 000
Please return this Home Link to school tomorrow.
For each question, circle all reasonable answers. (There may be more
than one reasonable answer.) All distances are as the crow flies. Be
sure to use the map scale on the next page.
1. About how many miles is it from New York to Los Angeles?
about 1,000 miles
more than the distance from Chicago to Dallas
about 2,400 miles
2. About how many miles is it from Chicago to Atlanta?
about 600 miles
more than the distance from Chicago to Seattle
Home Link 4 9
(Math Masters, pp. 110 and 111)
less than the distance from Chicago to Denver
INDEPENDENT
ACTIVITY
PROBLEM
PR
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VING
VIN
IN
ING
3. About how many miles is it from Seattle to Dallas?
about 2,600 miles
about 5,000 miles
more than the distance from New York to Chicago
4. About how many miles is it from New York to Atlanta?
Home Connection Children estimate distances between
U.S. cities on a map.
less than the distance from Denver to Atlanta
more than the distance from New York to Portland
about 750 miles
Math Masters, p. 110
3 Differentiation Options
READINESS
Exploring Scale
PARTNER
ACTIVITY
15–30 Min
To explore scale using concrete models, have children bring in toy
animals and compare them to the animals’ actual size. Children
bring a toy animal from home. They research the actual
dimensions of the animal—either on the Internet or in a book.
On large-size paper, children draw a rectangle for the animal’s
approximate size. They display their toy animals next to the
rectangle representing the animal’s actual size. If possible, provide
some models that are scaled up, for example, an insect that is
smaller than its scale model. Briefly discuss why toy animals are
usually scale models.
Teaching Master
Name
LESSON
49
䉬
Date
Time
Scale Drawings
Measure the length and width of your Math Journal in cm. Draw your
journal to scale on the grid. 1 cm represents 10 cm.
Scale
1 cm : 10 cm
Scale: 1 cm represents 10 cm
ENRICHMENT
Making Scale Drawings
PARTNER
ACTIVITY
5–15 Min
1. Math Journal
length: about
width: about
1
27ᎏ2ᎏ cm
1
21ᎏ2ᎏ cm
(Math Masters, p. 112)
To apply children’s understanding of scale, have them
make scale drawings of classroom objects. Children follow
directions on Math Masters, page 112.
Choose 1 more object. Record the length and width in cm. Draw the
object to scale on the cm grid.
Answers vary.
2. Object:
length: about
width: about
cm
cm
Math Masters, p. 112
Lesson 4 9
293