Rectangular Coordinate
Grids for Maps
Objectives To guide students in the use of letter-number pairs
and
ordered pairs of numbers to locate points on a grid; and to
a
provide practice using a map scale.
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Teaching the Lesson
Key Concepts and Skills
• Use a map scale. [Operations and Computation Goal 7]
• Estimate distances on a map. [Measurement and Reference Frames Goal 1]
• Use ordered number pairs to locate points
on a map. [Measurement and Reference Frames Goal 4]
• Use letter-number pairs to locate points
and regions on a map. [Measurement and Reference Frames Goal 4]
Key Activities
Students use letter-number pairs to find
locations on a map of Ireland. They use
ordered pairs of numbers to identify points,
give directions, and describe routes on a
campground map. Students use a map scale
to estimate distances.
Ongoing Assessment:
Recognizing Student Achievement
Use a Math Log or Exit Slip (Math
Masters, page 388 or 389). Family
Letters
Assessment
Management
Common
Core State
Standards
Curriculum
Focal Points
Ongoing Learning & Practice
1 2
4 3
Playing Angle Tangle
Student Reference Book, p. 230
Math Masters, p. 457
protractor straightedge
Students practice estimating
the measures of angles and
measuring angles.
Finding Real-Life Angle Measures
Math Journal 1, pp. 163A and 163B
straightedge
Students practice finding unknown
angle measures.
Math Boxes 6 8
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Moving on a Coordinate Grid
masking tape and index cards, or rope and
colored tape
Students describe the locations of points and
plot points on a life-size coordinate grid.
ENRICHMENT
Playing Grid Search
Student Reference Book, pp. 250 and 251
Math Masters, p. 486
Students practice naming and locating grid
squares and develop search strategies.
Math Journal 1, p. 160
Students practice and maintain skills
through Math Box problems.
Study Link 6 8
Math Masters, p. 194
Students practice and maintain skills
through Study Link activities.
EXTRA PRACTICE
Plotting and Naming Points
on a Coordinate Grid
Math Masters, p. 440
Students plot and name points on a
coordinate grid.
[Measurement and Reference Frames
Goal 4]
Key Vocabulary
index of locations letter-number pair ordered number pair map scale
Materials
Math Journal 1, pp. 161–163
Study Link 67
Math Masters, p. 388 or 389 (optional)
compass (optional) slate road atlas
(optional) ruler (optional)
Advance Preparation
For Part 1, you may want to use a road atlas of the United States to illustrate the use of indexes and
letter-number grid coordinates.
Teacher’s Reference Manual, Grades 4–6 pp. 249–252
Lesson 6 8
443
Mathematical Practices
SMP1, SMP2, SMP4, SMP5, SMP6, SMP8
Content Standards
Getting Started
4.OA.3, 4.MD.5a, 4.MD.5b, 4.MD.6, 4.MD.7
Mental Math and Reflexes
Math Message
Pose multiplication and division number stories.
Suggestions:
Turn to journal page 161. Find the city of Tralee on
the map. Be prepared to explain how you found it.
3 boxes of beads; 4 beads per box:
How many beads in all? 12
12 cups divided equally between 2 trays:
How many cups per tray? 6
Study Link 6 7 Follow-Up
8 boxes of cupcakes; 6 cupcakes per box:
How many cupcakes in all? 48
5 shirts per rack; 45 shirts in all: How many racks? 9
4 crates of tennis balls; 8 cans per crate; 3 balls per can:
How many tennis balls in all? 96
Students compare answers. The final problem is
more difficult because the angle is a reflex angle and
the half-circle protractor cannot directly measure
angles greater than 180°. Have students share their
strategies. One possible strategy: The smaller angle
has a measure of 60°. Subtract 60° from 360° to get
300°, which is the measure of the reflex angle.
180 seats in all; 10 seats per row; 6 rows per section: How
many sections? 3
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
(Math Journal 1, p. 161)
Social Studies Link The map on journal page 161 shows
the island of Ireland. Ask students how they found Tralee.
Make sure they know how to use the index of locations. Look
up Tralee in the list above the map. It is located in square B-2.
Find column B and row 2. The column and row overlap at a
square region. Tralee is located in this region.
NOTE The larger part of the island is the
Republic of Ireland. The smaller northern
part is Northern Ireland, a part of the United
Kingdom.
●
Student Page
Date
Time
LESSON
A Map of the Island of Ireland
68
Bantry
Belfast
Carlow
Castlebar
Derry
Dublin
Dundalk
Galway
Gort
Kilkee
B-1
F-7
E-3
B-6
E-8
A
0
50
Lahinch
Larne
Limerick
Mullingar
Navan
F-4
F-6
C-4
C-4
B-3
B
C
D
E
F
100 150 200 250 Kilometers
50
100
150
200
Donegal
Killybegs
Bangor
Belfast
Portadown
Downpatrick
Armagh
Sligo
Monaghan
Newcastle
Ballina
6
Newry
Castlebar
Cavan
Boyle
Westport
Drogheda
5
Mullingar
Ballinasloe
Galway
Naas
Bray
4
Port Laoise
Lahinch
Wicklow
Ennis
Roscrea
Carlow
Nenagh
Thurles
Kilkee
Rathkeale
Tipperary
Clonmel
New Ross
Wexford
Tralee
Fermoy
Millstreet
Killarney
Dungarvan
Cahersiveen Kenmare
Macroom
2
N
Youghal
Cork
W
Bantry
E
Kinsale
1
B
C
1
S
Clonakilty
Skibbereen
A
Ask students to look at the campground map and symbols on
journal page 162. Tell them that these are standard symbols for
United States topographic maps and most conventional road
maps. Students should notice that the map has been drawn on
a rectangular coordinate grid. Remind them that they can locate
points by naming ordered number pairs, such as (0,3) and (7,6).
Rosslare Harbor
Waterford
Mallow
Dingle
2
D
E
F
G
Math Journal 1, p. 161
EM3MJ1_G4_U06_137-169.indd 161
444
PROBLEM
BLEM
BLE
LEM
LE
L
EM
SOLVING
LVIN
VIN
VING
NG
3
Enniscorthy
Cashel
WHOLE-CLASS
ACTIVITY
(Math Journal 1, p. 162)
Arklow
Kilkenny
Limerick
Tarbert
Listowel
Locate Points on a Map
Dublin
Athlone
Loughrea
Gort
4
3
Using Ordered Pairs to
Longford
Tuam
Clifden
6
Dundalk
Navan
5
7
Dungannon
Omagh
Bundoran
Belmullet
Point out that in the Math Message students used letter-number
pairs to locate regions on a map. In the remainder of the lesson,
students will use ordered number pairs to locate specific points on
a campground map.
Larne
Ballybofey
7
Have students find other locations on the map.
8
Derry/
Londonderry Ballymena
Letterkenny
250 Miles
E-7
B-2
C-5
B-5
F-4
G
Coleraine
8
0
Omagh
Tralee
Tuam
Westport
Wicklow
B-4
F-7
C-3
E-5
E-5
Why is this letter-number pair method useful for finding
places on a map? It limits the search to a small part of the
map.
1/13/11 2:24 PM
Unit 6 Division; Map Reference Frames; Measures of Angles
To review the concept that the order of the numbers in an ordered
pair is important, ask students which ordered pair—(3,7) or
(7,3)—names a point in Blue Lake. To find point (3,7), start at 0,
go 3 to the right, and then go up 7. To find point (7,3), start at 0,
go 7 to the right, and then go up 3. The ordered pair (3,7) names
a point in Blue Lake; (7,3) does not.
Then ask students to name an ordered pair that describes the
location of the ranger station. Some students may name (6,1),
but point out that (5_12 , 1_12 ) is a better answer because the station
is located in the middle of a grid square.
Student Page
Date
Time
LESSON
A Campground Map
6 8
䉬
SCALE
N
144 145
0
0.2
0.4
0.6
0.8
W
1 km
E
S
12
County Road
11
S
10
Camping Area
Parking Lot
C
9
B
8
F
7
Picnic Area
Blue Lake
H
6
Fishing Dock
5
Practice naming ordered pairs with other examples, including
points that are between grid lines.
La
ke
Tra
il
4
3
Grave
l Roa
d
Canoe
Rental
2
1
Adjusting the Activity
0
A “ladder” metaphor may help some students plot ordered number
pairs. The first number tells where to put the ladder. The second tells how high
to climb. Or have students locate points by making heel-to-toe steps from the
origin (0,0) to named points (x,y). Students chant “over” as they move to the
x-coordinate and then “up” as they move to the y-coordinate.
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
Ranger Station
and House
0
1
2
3
4
5
6
7
8
9
10
11
Paved Road
Trail
Camping Area
Unpaved Road
River
Picnic Area
12
Math Journal 1, p. 162
V I S U A L
Math Log
or Exit Slip
Use a Math Log or an Exit Slip (Math Masters, page 388 or 389) to assess
students’ understanding of coordinate grid systems. Ask students to compare
the grid system on the map of Ireland to the one on the campground map.
Students are making adequate progress if they mention the following:
The grid of the map of Ireland identifies square regions using letter-number
pairs.
The grid of the campground map identifies points using ordered pairs of
numbers.
The order in a letter-number pair is not important, but the order of the
numbers in an ordered number pair is important.
[Measurement and Reference Frames Goal 4]
Student Page
Date
6 8
䉬
on a Map
(Math Journal 1, pp. 162 and 163)
Point out the map scale on journal page 162. To support English
language learners, write map scale on the board and explain that
the scale tells the relationship between the measured distance on
the map and the actual distance. Students should notice that each
side of a grid square represents 0.2 kilometers.
Sample answers:
Suppose you hiked along the lake trail from the
fishing dock to the parking lot. Estimate the distance
you hiked.
About
2.
The ranger made her hourly check. She started at the
ranger station. She drove northwest and then north
on Gravel Road to County Road. She turned east
onto County Road and drove past the parking lot
and the camping area. After she passed the canoe
rental, she turned right onto Gravel Road and drove
back to the ranger station. About what distance did
she drive?
About
3.
Estimate the distance around Blue Lake.
About
4.
You are planning to hike from the camping area to
the parking lot. You will stay on the roads or trails.
You want to hike at least 5 kilometers.
PARTNER
ACTIVITY
ELL
PROBLEM
PRO
PR
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
L
LV
VING
VIN
V
IN
ING
Finding Distances on a Map
Use the campground map on journal page 162 to complete the following:
1.
Estimating Distances
Time
LESSON
5.
a.
Plan your route. Then draw it on the map with a
colored pencil or crayon.
b.
Estimate the distance.
2–3
144 145
km
6.5–7.5
2.5–3.5
km
km
Answers
vary. km
About
Use the ordered number pairs to locate each item on the map. Mark a dot
at the location. Next to the dot, write the letter given for the feature.
Campground Features Chart
Location
Letter
parked car
(5,9)
C
boat
(3 ᎏ12ᎏ,8)
swing set
hikers
farmhouse
B
(8,11)
S
(10.5,6.5)
H
(ᎏ12ᎏ,7)
F
Math Journal 1, p. 163
Lesson 6 8
445
Ask students to estimate the distance along Lake Trail from the
fishing dock to the parking lot and to record it on journal page
163. Have students compare estimates and strategies:
Count the number of squares the trail passes through. 15 Each
side of a grid square represents a distance of 0.2 kilometers, so
the length of the trail is about 3 kilometers.
Make a mark on the edge of a sheet of paper. Place that mark
at the beginning point of the trail with the edge of the paper
along the trail. When the trail turns, make a mark on the edge
and turn the paper so the edge follows the trail. When you
reach the end of the trail, make a final mark. Measure the
distance between the first and last marks on the paper. Then
use the map scale to estimate the actual distance.
0 0.2 0.4 0.6 0.8 1 km
Use a traditional compass. On the map scale, set the opening of
the compass to represent 0.2 kilometers. Place the anchor point
at the fishing dock and swing the pencil point, marking off
about 0.2 km along the trail. Place the anchor point on this
mark and swing the pencil point, marking off the next 0.2 km
along the trail. Continue to the end of the trail. Count the
number of compass swings and estimate the distance. 11 or
12 compass swings, or about 2.2 to 2.4 km
Using a compass to estimate the distance around
an irregular shape
Ask: Why are the estimates obtained by the last two methods above
probably less than the actual length? Each segment on the paper’s
edge or each compass swing measures the path as though it were
made up of straight pieces. Such a path would be a little shorter
than the actual curved path.
Have partnerships complete the journal page and share results.
2 Ongoing Learning & Practice
Student Page
Date
LESSON
68
Time
Playing Angle Tangle
Dartboard Angles
A regulation dartboard is made up of 20 equal sectors. The measure of each sector’s
angle is 18 degrees.
5 20 1
12
(Student Reference Book, p. 230; Math Masters, p. 457)
18
9
4
14
Students play Angle Tangle to practice estimating and
measuring angles.
13
11
18°
6
8
16
10
15
7
19 3 17
2
Solve Problems 1–3 without using your protractor. After you have solved each
problem, write a number model with a letter for the unknown to show how you found
your answer.
1.
PARTNER
ACTIVITY
Adjusting the Activity
Draw ∠ABC around sectors 14, 9, 12, 5, 20, and 1. What is the measure of ∠ABC ?
12
5 20 1
18
9
4
14
A
13
11
19 3 17
A U D I T O R Y
6
B
8
16
7
Measure of ∠ABC =
Use this game variation as appropriate: Have students draw and
measure reflex angles. Their drawings should include the directional arc in the
appropriate place.
C
10
15
2
108 Sample answer:
Number model with unknown:
°
18 + 18 + 18 + 18 + 18 + 18 = m
Math Journal 1, p. 163A
137-169_EMCS_S_MJ1_G4_U06_576361.indd 163A
446
2/15/11 5:52 PM
Unit 6 Division; Map Reference Frames; Measures of Angles
K I N E S T H E T I C
T A C T I L E
V I S U A L
Student Page
Finding Real-Life Angle
INDEPENDENT
ACTIVITY
Date
Time
LESSON
Dartboard Angles
68
Measures
2.
(Math Journal 1, pp. 163A and 163B)
12
Math Boxes 6 8
(Math Journal 1, p. 160)
Study Link 6 8
4
13
F
11
8
16
D
3.
6
10
15
7
2
19 3 17
E
°
198 Sample
answer:
54 + a = 252
Number model with unknown:
Angle RST is a right angle. Use your straightedge to draw ray SM so that the measure of
∠MST = 54° and ∠RSM is an acute angle. Then find the measure of ∠RSM.
M
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 6-10. The skill in Problem 6
previews Unit 7 content.
Writing/Reasoning Have students write a response to the
following: Explain how to use the in and out numbers in
Problem 2 to determine the rule. Sample answer: The first
two rows have both the in and out numbers. The out numbers are
smaller than the in numbers, so I thought the rule might involve
subtraction. I subtracted the out number from the in number in
both rows and got 1.5 for an answer, so I knew the rule was –1.5.
18
14
Measure of reflex angle EFG =
G
5 20 1
9
Students practice finding unknown angle measures using real-life
examples. Using addition and subtraction, students find missing
measures on a dartboard.
INDEPENDENT
ACTIVITY
continued
The measure of angle DFE = 54° and the measure of angle DFG = 252°. What is the
measure of reflex angle EFG?
R
5 20 1
12
18
9
4
14
13
11
6
S
8
16
7
36
Measure of ∠RSM =
°
T
10
15
2
19 3 17
Sample answer:
90 - 54 = d
Number model with unknown:
Math Journal 1, p. 163B
137-169_EMCS_S_MJ1_G4_U06_576361.indd 163B
2/15/11 5:52 PM
INDEPENDENT
ACTIVITY
(Math Masters, p. 194)
Home Connection Students plot and label points on a
coordinate grid. They write the ordered number pair for
each point.
Student Page
Date
LESSON
68
B(
C(
D(
E(
2
4
1
2
4
5
4
3
2
1
,
,
,
,
,
)
5
A
)
D
2
0
in
-1.5
C
3
Complete the “What’s My Rule?” table
and state the rule.
Rule
B
E
1
)
2.
0
1
2
3
4
5
1.
out
Time
Coordinate Grids
68
4
)
Date
STUDY LINK
Name the ordered number pair for each
point plotted on the coordinate grid.
A(
Name
Math Boxes
1.
Study Link Master
Time
Plot and label each point on the coordinate grid.
A
(1,7)
B
(6,6)
144
10
3.6
2.1
10
8.5
C
(10,1)
7.2
5.7
D
(4,3)
6.4
4.9
E
(8,6)
F
(2,9)
F
9
8
A
7
B
6
4
)
144
H
D
3
2
G (9,1)
162–166
E
5
G
1
3.
∠EDF is
acute
(acute or obtuse).
4.
H
Cross off the names that do not belong
in the name-collection box below.
2.
32
E
D
Measure of ∠EDF =
80 °.
I
(
5
,
3
)
J
(
7
,
2
)
(5 ∗ 6) + 2
K
98 ÷ 3
L
10 + 15 + 7
M
93
142 143
Round to the nearest hundred-thousand.
149
a.
b.
c.
d.
6.
9,500,000
37,609,034 37,600,000
78,291,554 78,300,000
290,696,332 290,700,000
Fill in the missing fractions on the
number line.
O
9,540,234
P
0
14
12
3
4
Q
1
R
1
2
3
4
5
6
7
8
C
9
10
4 ,
7 ,
( 10 ,
( 1 ,
( 6 ,
( 8 ,
( 10 ,
( 3 ,
(
(
8 )
7 )
5 )
8 )
2 )
4 )
2 )
10 )
10
R
9
8
K
N
7
L
6
5
M
4
2
J
O
1
0
P
I
3
0
1
2
3
4
5
6
7
Q
8
9
10
Practice
3.
182 183
316
5.
Math Journal 1, p. 160
137-169_EMCS_S_MJ1_G4_U06_576361.indd 160
0
Write the ordered number pair for each point plotted on the coordinate grid.
N
5.
0
9∗4
81 - 49
F
(10,4)
28 ∗ 7 =
4,628
196
= 52 ∗ 89
4.
1,520
1,548 = 43 ∗ 36
304 ∗ 5 =
6.
Math Masters, p. 194
3/10/11 2:12 PM
EM3MM_G4_U06_177-202.indd 194
1/13/11 2:13 PM
Lesson 6 8
447
6
3 Differentiation Options
I
Love
Math
5
4
READINESS
Moving on a Coordinate Grid
3
2
SMALL-GROUP
ACTIVITY
15–30 Min
To provide experience locating ordered pairs, create or have
students help you create a life-size 6-by-6 coordinate grid on the
floor. Suggestions:
1
1
2
3
4
5
6
The book is located at blue 3, red 5.
Use masking tape to mark off the x-axis and y-axis on carpet
or a tile floor. Use index card “tents” (two colors) to mark the
points on each axis.
Tie two ropes together to create the x-axis and y-axis. Use two
colors of electrical tape at regular intervals to mark the points
on each axis. (See margin.)
Place objects on the coordinate grid. Have students describe
their location.
Give directions for students to follow to “plot” themselves on the
grid. For example, Walk over 3 blue steps. Now walk up 5 red steps.
PARTNER
ACTIVITY
ENRICHMENT
Playing Grid Search
15–30 Min
(Student Reference Book, pp. 250 and 251;
Math Masters, p. 486)
To apply students’ understanding of coordinate grids, have them
play Grid Search. Expect students to develop strategies for zeroing
in on the area where the queen has been hidden. They may also
invent variations—add more knights, add a king worth 7 points,
and so on.
Student Page
Games
Grid Search
Materials 䊐 1 sheet of Grid Search Grids for each player
(Math Masters, p. 486)
Players
2
Skill
Deduction; developing a search strategy
My Pieces
(Grid 1)
Object of the game To locate the opponent’s queen on a
coordinate grid in the fewest turns possible.
Plotting and Naming Points
INDEPENDENT
ACTIVITY
5–15 Min
on a Coordinate Grid
Directions
Players sit so that they cannot see what the other player is
doing. Each player uses 2 grids like those shown at the right.
Advance Preparation Before the start of the game, each player
secretly decides where to place a queen and 6 knights on their
Grid 1. They write the letter Q to record the location of the
queen and the letter K to record the location of each knight.
EXTRA PRACTICE
Opponent’s Pieces
(Grid 2)
(Math Masters, p. 440)
These are acceptable arrangements of the pieces:
To practice plotting and labeling points (including those between
grid lines) on a coordinate grid, have students complete Math
Masters, page 440. Fill in the page to create a new set of problems
for students each time they use the master or have students create
and solve their own problems.
These are not acceptable arrangements because the pieces
cannot be connected without skipping squares.
Planning Ahead
♦ The queen may be placed on any square.
♦ The knights may also be placed on any squares, as long
as the queen and the knights can all be connected without
skipping squares.
In Part 1 of Lesson 6-9, you will need a world globe and
a wall map.
Student Reference Book, p. 250
448
Unit 6 Division; Map Reference Frames; Measures of Angles