Session l
deqrees
Coordinatesand
Distanceson a Grid
rnTnu+€s
seconds
. deeima\ Sractb,
Whot Herppens
llolerlols
Students learn how to label points on a coordinate grid with an ordered
pair of positive numbers. They choose and describe addresses for their
own houses on a grid ofstreets and avenues. They practice using coordinates by delivering envelopeswith partial addressesto people living in
Grid City. Students learn the concept of distance on a grid as the lenglh of a
"taxicab path." Their work focuses on:
r StudentSheet1
(1 per student,and 1
transparency)
r StudentSheet2
(1 per pair)
r StudentSheet3
(1 per student,
homework)
r Transparencyof
Envelopes
r Family Ietter
(1 per student)
r Overheadprojector
r Coloredpensfor the
overheadtransparencies
r Determining locations on a map of Grid City
r Using coordinates to name and locate addresseson the Grid City map
r Writing coordinates as mathematicians do-that is, with horizontal coordinate first
r Finding "taxicab" distancesbetween two points on a grid
The chart below shows how students work during this session.
Whole
Class
40 min.
Creating Addresses in Grid CitY
Delivering Mail in Grid CitY
ftsading and Writing Coordinates as Mathematicians Do
Taxicab Paths in Grid CitY
Finding Distancesin Grid CitY
Show the Grid City map (Student Sheet 1) on a transparencyFretend we all moved to Grid city. Here is a map of the city, with a
school, a swimming pool, and each of your houses. In Grid city, each
building is at a corner, or intersection, of two roads. One road is a street
lindicate the streets along the horizontal arisl and one road is an avenue
lindicate the avenues along the uertical axisl.
Creating
Addressesin
Grid City
Each of you has to choose a house. I'll label that house with your initials-
Session 1: Coordinates and Distances on a Grid s 5
Ask two to four student volunteers to choose a house and tell you which
one it is. Let the students figure out how they might do this without pointing (you may want to face the screen so students have to use words).
Otld Cltt
As each student properly locates a house, write the student's initials near
that house- The usefulnessof having a standard way to refer to the
addresses(by convention, a number for the street followed by a number for
the avenues)should emerge from this discussion.
ot234567E9tOtt12!3ta
9. 5r- 9, 5r- 9. S'. 9. t. 9- 9. 5r. 9. 9. 5r- t
Delivering MaiI
in Grid Citv
15 t6
5r. 9,
Give studentscopiesof the Grid City map on StudentSheet1. Have them
find and Iabel the houses that have already been chosen.Ask the rest ofthe
students to choose a house by giving the two-number address (for example,
'Eighth
Street and First Avenue"). After each student states an address, all
students write on their sheetsthat student's initials next to the corresponding house while you label your map on the overhead. If there are more students than houses, students can draw a new house at each location the
final students choose.
Show the transparencyofEnvelopes(p. 148), or draw the envelopeson the
chalkboard.
The mail carrier in Grid City has several envelopes that have been
smudged. Can you help lill in the missing information? For example, on
the first envelope, can you tell whose initials have been smudged?
Look at the second envelope. What can you figure out about where this
envelope should go? What, tben, could the missing information be? Are
there any other possibilities?
The rest of the sheet contains envelopes with no address and no initials. Use
these for students to make up smudged envelopes to challenge one another.
For the first few envelopes, have one student suggest what to fill in for two
of the three pieces of missing information-initials, street number, and
avenue number-and write these on the overhead or on the board- Then
have the rest of the class identify the missing third piece of information.
6 t Incestigation1: I-ocatingHousesand Shipson a Grid.
For the last few envelopes, ask a student to suggest what to fill in for only
one of the three pieces of information. Ask the other students to fill in the
other two missing pieces. Is there more than one possibility? How do you
know?
The Grid City mail carrier was told to deliver an interesting-looking
package to 8th and 1Oth. Whose house should that package be delivered
to?
Accept a few responsesfrom students, enough to indicate that the phrase
"8th and 10th" could be interpreted in more than one way, as either 8th
Street and 1Oth Avenue or 8th Avenue and 1Oth Street. Tell students that
the mail carriers decided to adopt the rule that mathematicians use when
they are locating a particular point on a grid like this one. They decided
always to say the street (along the horizontal) first and the avenue (along
the vertical) second.Therefore, "8th and 10th" means 8th Street and 1Oth
Avenue.
Reading and
Writing
Coordinatesas
Mathematicians
Do
Who lives at 8th and 1Oth?At 1Oth and 8th?
Suggest a few other locations and have students show the locations on the
overhead of Grid City. When students are comfortable locating houses on
the grid this way, introduce the written notation for labeling coordinates.
When mathematicians are locating points on a grid, they write down the
numbers like this: (8, 10) lwrite the ordered pair of numbers on the boardl.
This notation for writing two numbers in a specific order is called an
ordered pair. The numbers in the ordered pair are called the coordinates of tbe point.
Coordinates give an exact location ofa point. The first coordinate, or
number in the ordered pair, is always on the horizontal number line
going from left to right. The second coordinate, or number in the
ordered pair, is always on the vertical number line.
Let's try a few.
Write down the coordinates of each house on the grid so students can see
how the written notation corresponds to the location of the points on the
grid. You may wanl to have students come up to the overhead of Grid City
and show how they find particular points, Give a few clues, such as the following:
Who lives at (4,8)? How do you know? Where is DK's Isubstitute the initials of someonein your classl house? Where would a house be whose
coordinates are {0, 6}?
Session l:Coordinates and Distances on a Grid .
7
when
Then have students take turns glving clues to the rest of the class.
point to a
students seem comfortable with locating points on the grid'
natcoordinates
word
the
Use
house and have them write the coordinates.
to
begin
will
students
so
urally as it fits into this and subsequent sessions
use it appropriatelY.
Taxicab Paths
in Grid City
Show the Grid City map transparency'
only
Pretend you drive a taxicab in Grid city. You can drive your taxi
the
across
not
along the lines representing the streets and the avenues,
,q'u'*"betweenthe|ines.Also,youggnnotdrivealongthesame|ine
Let's see
more than once, since the passenger would have to pay double'
call the
we'll
pool.
how far it is by taxi from the school to the swimming
blocks
distance between two points on the gnd a "block.- How many
pool?
does the taxi have to drive from the school to the swimming
Some students may have trouble understanding the unit of measurement
blocks'
used here. A few may start out counting intersections rather than
See
blocks.
of
number
the
arriving at an answer that is one greater than
stuhelp
(pto
10), for ways
the Teacher Note, what Are we Measuring?
dents focus on blocks,
pool and
Have a student choose one path from the school to the swimming
path on the
give the coordinates ofeach intersection as you draw his or her
on the
number
the
record
overhead. count the blocks the taxi traveled and
board.
Are there other paths the taxi could follow?
the
students show different paths between the two points representing
go over the
school and the swimming pool. Be sure not to allow paths that
path,
draw
for
each
color
different
same block more than once. using a
After
board.
the
on
each path on the overhead. Record the path lenglhs
shortest
the
students have offered two or three paths, ask them to consider
path.
pool'
Now try to find the shortest path from the school to the swimming
ls there more than one shortest path?
the
The goal of this discussion is to reinforce the definition of distances on
paths
shortest
grid and to come to the conclusion that there are many
between two Points.
8
t Inuestigation 1: Locating Houses and Ships on a Grid
A final guestion you may want to ask about this journey is this:
What if the taxi starts at the swimning pool rather than at the school?
What is the shortest path? Is it the same length as the path from the
school to the swimming pool.
Pairs work together on the Grid City maps (Student Sheet 1), answering the
questions on Student Sheet 2, Finding Distances in Grid City. \A/henthey are
finished, briefly discusstheir answers to be sure the concept of distance is
clear. Students will do more work with distances on the grid in the next
session.
Finding
Distancesin
Grid City
Session I Follow-Up
Coordinate Mystery Send home the family letler or the Inaestigations al
Home booklet, along with a copy of Student Sheet 3, Coordinate Mystery.
Students complete the picture at home.
fif
Honework
Up until now, students have focused on measuring distances along vertical
and horizontal lines on the grid. Drawing the Coordinate Mystery picture
requires drawing lines diagonally across squares. Let students know that
on this student sheet, and when they make up their own coordinate mysteries, they can draw lines that are not on grid lines.
Session 1: Coordinatesand Distances on a Grid t
9
Teacher Note
Whaj Are We Measuring?
ln this session,we are measuring distances-the
number of blocks, or line segments,along the
path.
Here the path between the two housesis
6 blockslong.
Some students, though, may initially think of a
"block" as a square region.
Others may count "blocks" by counting each
point, or intersection, with the starting point
countedas "1."
I O r
Inuestigation 1: Locattng Housesand Ships on a Grid
You may find that having students think in terms
of motion, visualizing themselves walking along
the roads in the diaglam, helps them think
about the problem in ways consistentwith our
definition of "block'" You may want some students to act out being a taxi, counting blocks as
they drive along streets and avenues'
Grid GifY
lrr
I 6 Ave.
1
I 5 Ave.
Ir 5r
P
| 4 Ave.
IIJ
.t
gt
,t
A.
trl
5t
l
tr
+
\
l T
v
v
^
i
4
| 3 Ave.
IIl
l
IIl
3l
t
.t
| 2 Ave.
lll
\
,
$
A
ln
I I Ave.
J
I O Ave.
^
/
t\
ttl
ut
l
Ir -g}r
IIJ
pc rl
-3,
9 Ave.
8 Ave.
lr
IIJ
sr
I
l\
1Bl
2
lrJ -5'
-51
*
7 Ave.
t
6 Ave.
IIl
I
|I,
-El
II
5l
frtl ri
5 Ave.
A,
tBl
scl ool
t
4 Ave.
3 Ave.
3t
IIJ
,*.
A.
lrrt
-51
./
tr
a
*
!t
Il'
t-ug
-5'
IB
.:l1
Ill
.cl
!/
/
la, -91
2 Ave.
O Ave.
a
A
I Ave.
lfJ
lll
5'
5t
i
2 3 4 5 6 78
I
5t. 51. st. 51. 5f. 51. 5t. st.
o
O Peorson Educotion, Inc
r35
9 I O I I r2 t3 r4 r5 t6
si. sl. 51. 5t. 51. 51. St. 51. 5t.
Investigation I 'Session I
Suniin SiiPs and Grid Patterns
Finding Distcnces in @rid CifY
1. What are the coordinatesof your house?
The coordinatesof the schoolare (7, 5).
How many blocksis your housefrom the school?
2. Who is your Partner?
What are the coordinatesof your partner's house?
How many blocks is your housefrorn your partner'shouse?
3. \,Vhosehouseis closestto yours?
How many blocks is your housefrom that house?
4. Whosehouseis farthest from yours?
How many blocks is that housefrom your house?
5. How many housesare lessthan 8 blocks from the school?
ryVhosehouses are theY?
6. Who lives more than 16 blocks from the school?
@ Pmrrcn Edrotion,
Inc
t36
Investigation 1' Session1
Sunken Ships arut Cria fatterns
Dofe
Coordinote MYslerY
What is the mystery dot-to-dotpicture?
Draw it by connectingthe coordinates'
Start at this Point:
(9,6)
to each of these
Then draw straight line segTnents
pointsin order:
(1, 6) (8, 1) (5, 9) (2, 1) (9, 6)
Your lines can go diagonally on the grid'
to
o
@ Peorrn
Edrotion,
lnc-
o1234567891o
137
Inuestigation 1' SessionI
Sunken SniPs-onaCrid Pauerns
ENYEIOPES
12m St. Ave.
12tnst. 11thAve.
@ Pmrson Educotion. Inc
sr.
Ave.
st.
Ave.
st.
Ave.
sr.
Ave.
st.
Ave.
St.
Ave.
ta8
Investigation I' Resource
Sunfen Shtpsand Crtd Patterns