Hands-On Math Lab

by Fran Endicott Armstrong, Ph.D.

Graphing relationships involving whole numbers on a first-quadrant rectangular coordinate (Cartesian coordinate) graph can reveal interesting properties of arithmetic operations.

A two-dimensional rectangular coordinate system essentially consists of two number lines intersecting perpendicularly at their 0s. Each number line then is called an axis (plural: axes). So there is a horizontal axis and a vertical axis. The point of intersection of the two axes at their 0s is called "the origin." Each axis has a particular unit value which need not be the same for both number lines. In our discussion we will use a unit value of 1 on each axis. Then there is a one-to-one relationship between every point in the plane and an ordered pair of numbers. We agree that the first number (coordinate or component) in the ordered pair represents horizontal distance from the origin. And the second component of the ordered pair represents vertical distance from the origin.

 Horizontal Vertical sideways up or down

The convention is that the positive direction on the horizontal axis is to the right and the positive direction on the vertical axis is up; this is usually indicated by putting an arrowhead on the right end of that portion of the horizontal axis which is drawn and an arrowhead at the top of the portion of the vertical axis which is drawn. (Each axis goes on infinitely, but we only draw a finite portion of the axis.)

Many schools have a floor tiled with square tiles which are typically either 9 or 12 inches on a side. Such a floor makes a great place to have a rectangular coordinate system on which children can walk. The teacher may want to show the axes with colored tape or draw the axes on the tiles with washable or permanent marker. A 12 by 12 grid is convenient. Then the children need to practice walking to the position indicated by a given ordered pair, for example (5, 3) by starting at the origin and walking 5 units horizontally (sideways to the right as they face up the vertical axis) and then walk 3 units vertically or up. Don't forget to include among the ordered pairs ones that have 0 as the first component (no sideways movement, so the point is on the vertical axis) and 0 as the second component (no vertical movement, so the point is on the horizontal axis). In practicing plotting points on the rectangular coordinate system on the floor, either the students may stand on their points or you may want to use little plastic figurines of characters (from Sesame Street or from Disney movies or Berenstain bears or what-have-you). "Caroline, put Pocohantas on (3, 7)." (When reading ordered pairs, refrain from saying "and" between the two numbers. Instead, just pause at the comma.)

GRAPH HIDE & SEEK

(a cooperative game developed by Fran Endicott Armstrong
from an idea created by Debbie Char and her 9-year-old son Justin on 12-3-97)

Materials

A 12 by 12 rectangular coordinate grid on the floor (or a smaller
floor grid if you prefer)

169 paper or plastic cups if using a 12 by 12 grid (121 cups for a 10 by 10 grid, 36 cups for a 5 by 5 grid). In general, (n + 1)2 cups for an n by n grid)

Some trinkets or figurines or wrapped candy pieces to hide under the cups on the floor grid

How to Play

Have student volunteers carefully place one cup on each node of the grid (A node is the corner of the square where the horizontal and vertical lines intersect). Don't forget that there are nodes on the horizontal and vertical axes themselves.

After the cups are placed on the nodes, the teacher starts the game by directing the students to line up and close their eyes ("No peeking."). The teacher is the first Hider. The Hider hides an object (a trinket or a figurine or a treat, e.g., Big Bird) under one of the cups and writes on the board the ordered pair (e.g., (3, 5) ) representing the node where the object is hidden. Then the Hider says "Ready" which signals the students that they may open their eyes.

The Hider points to the ordered pair written on the board and announces to the class, "Big Bird is at (3, 5)." (When reading ordered pairs, refrain from saying "and" between the two numbers. Instead, just pause at the comma.)
The first Seeker tries to find "Big Bird." The Seeker gets one unaided chance to pick up a cup to find the hidden object. If the Seeker lifts up the cup at the correct location and thus finds the object, then the Seeker replaces the cup at that node, shows the class the object while announcing "Big Bird was at (3, 5)," and returns the object to the container of objects (or keeps the object if the teacher has decided to give away the objects this particular time the class plays Graph Hide & Seek. The teacher may want to do that for a special holiday celebration.).

There are two possible error situations:

1.) If the Seeker picks up a cup at an incorrect location on the first try and feels that s/he needs help finding the correct location, then the Seeker calls out "Conference." Students raise their hands if they want to confer with the player by giving directions to the correct location identified by the ordered pair on the board. The Seeker picks one of the students with a raised hand to be his/her Conferee. The Seeker goes to the origin and the Conferee gives verbal directions (preferably) or hand signals indicating how the Seeker should move to get to the right location (e.g., "To get to (3, 5) from the origin, move 3 units to the right and then go up 5 units."). The Seeker then arrives at the correct location, finds the object, announces "Big Bird was at (3, 5)," and returns the object to the container of objects (or keeps the object if the teacher is giving the objects away this day).

2.) If the Seeker picks up a cup at what s/he believes to be the correct location, but the object is not there, and the Seeker still thinks s/he chose the correct location, then the Seeker says "Conference with the Hider." The Seeker and the Hider confer. Either the Seeker convinces the Hider that s/he has written the ordered pair incorrectly or the Hider convinces the Seeker that s/he has incorrectly identified the location given by the ordered pair written on the board. Whichever player needs to corrects his/her mistake does so. That is, either the Hider corrects the ordered pair written on the board and announces the corrected location of the object or the Seeker uses the information provided by the ordered pair to correctly locate the node at which the object is hidden. Then the Seeker goes to the location indicated by the ordered pair, finds the object, and returns the object to the container of objects (or keeps the object if the teacher is giving the objects away this day).

Then the Seeker becomes the new Hider. While the rest of the students close their eyes, s/he picks another object from the container of objects (e.g., Elmo), hides it under a cup, and writes on the board the ordered pair (e.g., (0, 9) ) representing the location where the object is hidden. Then the Hider says "Ready" indicating that the rest of the students may open their eyes. The Hider announces, while pointing to the ordered pair written on the board, "Elmo is at (0, 9)." Then the next student in line becomes the new Seeker and tries to find Elmo at (0, 9). Play continues in this manner until each student has had a turn to be a Seeker and a Hider (The last Hider hides the object for the teacher, who is the last Seeker).

Other than "conferences," this game should be played in silence and without signaling.

COLLINEAR GAME

(a cooperative game devised by Fran Endicott Armstrong 12-2-1997)

Materials

A 12 by 12 rectangular coordinate grid (preferably on the floor, but one projected onto a wall with an overhead projector or drawn on a chalkboard or whiteboard would work.)

Game pieces for the players to place on the grid

Two bowls, one labeled "Horizontal" and the other labeled "Vertical," each containing pieces labeled with the whole numbers 0 through 12

An adequately long straightedge for checking whether the pieces are collinear (for the floor grid, a long piece of quarterround wood molding or light metal or plastic strip)

How to Play

Players take turns putting a character figurine or a game piece on the rectangular coordinate grid. The goal of the game is to correctly place pieces on the grid and to recognize when three pieces are collinear (i.e., lie on a straight line). [Any two pieces are automatically collinear, since two points determine a line. Thus the interesting situation is when three pieces are collinear.] To determine the location of each piece, ordered pairs are created by drawing at random (with replacement) a number from a bowl labeled "horizontal" and then a number from a bowl labeled "vertical." The teacher or a student could write the ordered pair on the chalkboard. The game is to be played in silence with no coaching, groaning, or nonverbal signals.

There are three exceptions to the rule of silence:
1.) The current player may call out "Conference" when s/he wants to confer with another player as to how to correctly place the piece. Students raise their hands if they want to help the player. The player picks one of the students with a raised hand to provide direction. The player goes to the origin, and the student conferee gives verbal directions (preferably) or hand signals indicating how the player should move to get to the right location (e.g., "To get to (3, 5) from the origin, move 3 units to the right and then go up 5 units.").

2.) A player other than the current player may call out "Conference" when s/he questions whether the current player has correctly placed the piece and wants to discuss it with the current player. Following the conference the current player may move the piece to another location if s/he thinks that would be the correct placement.

3.) Any player calls out "Three collinear" or ("Three in a line") if h/she thinks that there are now three pieces that are collinear, that is, lying on a straight line.

Only the current player may move around the outside of the grid area to see if any pieces are collinear. The rest of the players must do the best they can from where they are (to prevent the chaos of many students running around the room).

After a player calls out "Three collinear," then the player must identify by their ordered pairs the three points believed to be collinear. Then the straightedge is used to check.

If desired, the game can continue to be played after "Three collinear" with players now looking for Four Collinear.

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Updated 12/1/99