Coordinate Geometry for Third Graders Author(s): Robert F. Smith Source: The Arithmetic Teacher, Vol. 33, No. 8 (April 1986), pp. 6-11 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/41194163 . Accessed: 14/06/2014 15:29 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Arithmetic Teacher. http://www.jstor.org This content downloaded from 91.229.229.205 on Sat, 14 Jun 2014 15:29:01 PM All use subject to JSTOR Terms and Conditions
Transcript
Coordinate Geometry for Third GradersAuthor(s): Robert F. SmithSource: The Arithmetic Teacher, Vol. 33, No. 8 (April 1986), pp. 6-11Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194163 .
Accessed: 14/06/2014 15:29
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp
.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].
.
National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.
http://www.jstor.org
This content downloaded from 91.229.229.205 on Sat, 14 Jun 2014 15:29:01 PMAll use subject to JSTOR Terms and Conditions
Let's Do It- activity approaches to teaching and learning mathematics, with extensions through grade six.
Few educators would question the validity of the following statement: "The specialized knowledge and skills needed to deal with maps and globes - specifically, latitude, longi- tude, and scale - are founded in math- ematics, not in social studies, and should be treated accordingly" (Wel- ton and Mallon 1981, 286). Prepara- tory to acquiring concepts involved in latitudinal and longitudinal coordi- nates, children should understand (1) the need for a frame of reference, that is, a horizontal and vertical axis; and (2) that specific locations in space can only be identified by both their hori- zontal and vertical positions with re- spect to that frame of reference. Clearly, such understanding is nur- tured through the development of children's ability to name and locate coordinates. Since many social stud- ies programs introduce the study of latitude and longitude in fourth, fifth, or sixth grade, it is logical for mathe- matics programs to introduce devel- opmental work on coordinate geome- try at least by the end of third grade.
A problem arises when one exam- ines carefully how coordinates or the
An associate professor in the School of Educa- tion at Brooklyn College, Brooklyn, NY 11210, Robert F. Smith teaches graduate and under- graduate methods courses in early childhood education.
6
graphing of ordered pairs is intro- duced in a majority of mathematics textbooks. The topic appears sud- denly; neither the teacher's edition nor the students' text provides devel- opmental activities to promote stu- dents' conceptual understanding of coordinates and to develop their abil- ity to name and locate coordinates on a graph. In any one mathematics text series, coordinate geometry or graph- ing of ordered pairs may be allocated three pages in the third grade, five pages in the fourth grade, and five to six pages in the fifth and sixth grades. Such a sparse treatment of an impor- tant topic, spread out over four years, can hardly be said to promote either an understanding of coordinate geom- etry or skill in naming and locating ordered pairs on a graph.
A Developmental Unit Because of the lack of carefully se- quenced instruction for developing students' ability to graph ordered pairs, the introduction of a develop- mental unit on coordinate geometry in third grade seems appropriate. As a consultant for a program of mathe- matics enrichment in a public school in Brooklyn, New York, I developed and introduced such a unit to a third- grade class. Activities in the unit make use of geoboards, dot paper, maps, and graph paper.
Introductory activities with a geoboard These activities are used for motiva- tion as well as for the development of simple skills in locating and naming
points on a geoboard. They are pre- requisites to the subsequent activities. Children should have had some previ- ous experience with geoboards: mak- ing shapes and designs with one or more rubber bands, copying shapes or patterns, making triangles and squares of different sizes, and so forth. If children have not had such experience, they should at least have had the opportunity to explore the geoboard so that they understand that it has five pins (points) across and five pins (points) down, or twenty-five pins in a 5 x 5 array.
1. "Name that point**
In this activity the teacher holds the geoboard so that the array of pins is not visible to the children. The teacher then covers one of the pins and asks the children if they can guess the point. Various responses are elic- ited from the children, but it soon becomes obvious to them that a sys- tem for describing the location of each
Arithmetic Teacher
Fig. 1 A grid for "name that point"
This content downloaded from 91.229.229.205 on Sat, 14 Jun 2014 15:29:01 PMAll use subject to JSTOR Terms and Conditions
Fig. 2 Points are labeled with a number for the column and a num- ber for the row.
Row 5 • • • • •
4 • • • • •
3 • • • • •
2 • • • • •
1 • • ♦ • ♦ T- C'l CO 4t LO
с
i point is necessary to "name that point/'
The teacher should allow the chil- dren to be creative in their sugges- tions for labeling the points. The teacher may ask them if it would help if the points or pins were labeled with letters, for example, Л, B, and C. An appropriate diagram can be drawn on the chalkboard or shown on an over- head projector (fig. 1). The children can now refer to this labeled geoboard to "name that point." Children should play the game in groups of two or more for additional practice in naming and locating points. Circles of colored paper with sticky backs can be lettered and placed below each pin.
2. "Name that point" - a different way
In the next activity, the class tries to think of another way to describe or label the points on the geoboard. One of the children in my third-grade class came up with the following sugges- tion: "We can find the point by telling what row it is in." This was an excel- lent idea though only a partial solu- tion, as the children soon discovered. They experimented with naming only the row for a few minutes, but quickly realized that just naming the row was not enough. Several children them- selves now suggested that "we also have to name the column the point is in." This is exactly what I was leading
April 1986
up to - naming a point by describing its position by both the column and the row. Some children had difficulty distinguishing between columns and rows, so reinforcement of this distinc- tion was necessary. An appropriate diagram (fig. 2) can be displayed to help the children develop skills in lo- cating points in terms of columns and rows (Trivett 1971). // is essential that the children understand that they must first name the column, then the row, in which the point is located. The teacher might use a mnemonic device, such as "C comes before R in the alphabet, so column comes before row."
Other suggested activities include the following:
1. The teacher makes a hidden tri- angle or square on a geoboard and then names the column and row of each point that represents a corner of the shape (fig. 3). The children locate these points on their geoboards, con- nect them with rubber bands, and then compare their shapes to the model.
2. The children make their own hid- den shapes on the geoboards. Several children are called on to describe their shapes by just naming the column and row of each corner point. The rest of the class tries to make the identical shape. This activity can also be com- pleted by children working as part- ners, taking turns making and naming
each other's shape.
Geoboard dot-paper activities These activities allow children to make a smoth transition from naming and locating points on the geoboard to naming and locating points on graph paper. Geoboard dot-paper activities are an important link between hands- on work with a geoboard and paper- and-pencil work with graph paper.
When introducing the dot paper, the teacher should emphasize that children must first label their columns and rows (fig. 4). The following are a few suggested dot-paper activities:
1 . Copy my shape
Children can work as partners in drawing and naming shapes on dot paper. One child draws a shape and names the column and row of each corner point of the shape; the partner has to draw that exact shape on his or her dot paper.
2. Tac-tac-toe and Battleship
Familiar games like these can be used to give children practice in naming and locating points. Using a diagram of a 5 x 5 array on an overhead transparency or the chalkboard, the class can play Tic-tac-toe. The class can be divided into an X team and an О team. As children from each team are called on to name a point - they
7
Fig. 3 Students create a figure using coordinates given by the teacher.
Fig. 4 Geoboard dot paper
This content downloaded from 91.229.229.205 on Sat, 14 Jun 2014 15:29:01 PMAll use subject to JSTOR Terms and Conditions
must give its column and row - the teacher or student places an X or an O, respectively, on the array (fig. 5). The first team to get three Xs or Os vertically, horizontally, or diagonally is the winner.
Children can also play Battleship as partners; each child draws a fleet of ships on dot paper in any way. For example, a battleship can be a 2 x 1 rectangle, an aircraft carrier can be a 3 x 1 rectangle, a destroyer al x I square, or a gunboat a 1 x 0 line segment. At the beginning of the game, each child draws the same number of ships. Of course, each "fleet" is hidden from the partner's view. The purpose is to sink the op- ponent's ships by locating their posi- tion on the dot paper. To do so, one child names at least one set of coordi- nates (the column and row) of one of the points of the ship's location. If the child names such a point, the oppo- nent's ship is sunk. The first player to sink all the opposing ships is the win- ner.
3. Introducing the "0" coordinates
No reference has yet been made to the horizontal and vertical axes that rep- resent the "0" coordinates. A smooth transition from locating coordinates on dot paper to locating points in quadrant I on graph paper requires the introduction of these "0" coordi- nates. To do so, draw the diagram in figure 6 on the chalkboard or on an overhead transparency; this diagram represents a 5 x 5 array with all lines labeled except the "0" axes.
The teacher or student indicates a "point" and asks someone to name its column and row. After several points are identified, the teacher can then place a finger on a point on the verti- cal axis, (0, 3), for example, and ask the children if they can name its posi- tion (column and row). Suggestions are elicited as to how they can name this point. If they have difficulty sug- gesting a name, the teacher can refer to the fact that the diagram has both a horizontal and a vertical number line. Third graders are familiar with num- ber lines and the fact that they usually begin with 0. The third graders partic-
8
ipating in the pilot of this developmen- tal unit had no difficulty understand- ing the function of the 0 in coordinates of a particular point.
The activities outlined earlier (e.g., naming of shapes, tic-tac-toe, or Bat- tleship) can now be extended to give children practice in recognizing and naming coordinates using 0. Horizon- tal and vertical lines can be numbered beyond 5 to extend graphing opportu- nities. Additional activities using only one quadrant of a graph can be com- pleted for further practice, reinforce- ment, and fun (worksheet 1).
Map-reading activities These activities allow students to ap- ply their newly learned skills to map- reading activities typical of those found in third-grade social studies texts. Children can use their new skills in practical situations concern- ing the location of points of interest in the community, the provision of di- rections for traveling, and so on. In addition, such activities enable the teacher to integrate the subject matter in a meaningful way.
Simple grids using maps of neigh- borhoods or communities are usually presented first (worksheet 2(a)). Chil- dren can be asked to name the coor- dinates of the school, the library, a local movie theater, a park, or a friend's house. In addition to doing
map activities in standard texts, both the teacher and the children can make maps of their own community or even design their own model community.
When the children are ready, the teacher can introduce maps that rep- resent larger areas and include more detail (worksheet 2(b)). Many such maps use letters and numbers to rep- resent horizontal and vertical lines on a grid. However, children should not have difficulty in relating to this dif- ferent way of naming coordinates, since the basics of using horizontal and vertical frames of reference have already been established. The use of these maps depends on the teacher's ingenuity. Children can match coordi- nates to points of interest; coordinates can be identified and students re- quired to name what they find; or cutouts or pictures of objects can be placed on specific coordinates.
Coordinate geometry on graph paper These activities focus on strengthen- ing children's understanding of coor- dinates while developing their skills in naming and locating coordinates in two or more quadrants. Using a dia- gram of quadrant I on the chalkboard as a frame of reference, the teacher puts a point to the left of quadrant I and asks the children if they can name that point (fig. 7). This question is a
Arithmetic Teacher
Fig. 5 Coordinates are given for points in tic-tac-toe.
Fig. 6 A grid on graph paper for students to discover the need for coordinates with 0
5 f t 1 1 1
4 - ( i^-h- h и
3 ,, ..- н 1- н
2 >- -ni и i - h
m 1 2 3 4 5
This content downloaded from 91.229.229.205 on Sat, 14 Jun 2014 15:29:01 PMAll use subject to JSTOR Terms and Conditions
Use the cut-outs to build Space Center Juniper. 1 . Put the launch pad at (5, 4). 2. Put the control station at (1 , 2). 3. Put the waiting station at (2, 5). 4. Put the astronaut at (4, 5). 5. Put the rocket transporter at (3, 1).
I I /V I ■ jfc"íí I i i >^" » i
Waiting Station Launch Pad Control Station Rocket Transporter Astronaut
From the Arithmetic Teacher, April 1986
challenge to the students, and they enjoy guessing the name of a point in such unfamiliar territory. The teacher encourages the suggestion that the horizontal number line can be ex-
Apr// 1986
tended from 0 to the left using nega- tive numbers. Such encouragement includes examples, with concrete il- lustrations if necessary, of tempera- tures below zero or geographic loca-
tions below sea level. When the horizontal axis has been
extended to include negative numbers on the chalkboard diagram, the chil- dren can be given individual copies of
9
This content downloaded from 91.229.229.205 on Sat, 14 Jun 2014 15:29:01 PMAll use subject to JSTOR Terms and Conditions
Fig. 7 What are the coordinates of the point on the left?
Fig. 8 Students complete the grid.
the diagram and asked to complete it by drawing in the necessary horizon- tal and vertical lines (fig. 8). Once the children have completed their own graph (two quadrants), they should now be able to name the point intro- duced earlier.
Having the children complete quad- rant II on their own provides a logical transition to their use of graph paper. They need to understand, however, that first they must identify the verti- cal and horizontal axes of their graphs by darkening horizontal and vertical lines in the center of the paper and labeling the intersection "0." Then they can number as many horizontal and vertical lines as they wish. The activities described earlier can be used to extend and reinforce the children's ability to name and locate coordinates in quadrants I and II.
The introduction of coordinates in quadrants III and IV is relatively easy. The only new understanding re- quired of students is that the vertical axis must now be labeled with nega- tive numbers as well. My third grad- ers had no difficulty making this sug- gestion themselves to name a point in either quadrant III or IV. Once they understood how to name coordinates in these quadrants, they eagerly en-
April 1986
gaged in familiar activities that strengthened their understanding and skill.
Exactly how and when the term coordinate should be introduced to younger children is at the teacher's discretion. Yet, once the children un- derstand that they need to identify both a column (horizontal) and row (vertical) to name or locate a specific point on a graph, the term coordinate becomes more meaningful. Other terms, such as axis or quadrant, are probably more difficult for a third grader to understand or remember, and their introduction during this unit is not necessary.
Conclusion Most third graders who participated in this developmental unit were able to name and locate points in all four quadrants of a coordinate plane by the end of the unit. They understood not only the need for a frame of reference, that is, horizontal and vertical axes, but also that specific locations in a plane can only be identified by both their horizontal and vertical positions in that frame of reference.
The introduction of a unit on coor- dinate geometry that includes a pro-
gression of simple to more complex activities is possible at an earlier grade level than normally suggested. From a developmental point of view, third graders are able to understand the concepts and master the skills required. From a pedagogical point of view, it is more logical to present coordinate geometry as an integrated unit rather than in a piecemeal fash- ion.
Even though third graders will probably not encounter the specific concepts of latitude and longitude un- til a later grade, their participation in this developmental, integrative ap- proach to the mathematics of coordi- nates will equip them with the skills required to use maps and globes in subsequent social studies courses.
Bibliography Butzow, John W. "Y Is for Yacht Race."
