Coordinate graphs use ordered pairs of numbers to designate a single point on a grid that represents two variables at once. The data from any graph students make can be displayed using ordered pairs of numbers on a grid, but not all data is usefully displayed in such a manner. Only data that can be used to make specific predictions for future events is best represented and interpreted in coordinate graphs. Why this is so will can be made clear to students. First they need to learn the basic techniques of assembling coordinate graphs. Once these techniques are learned, they can be guided through experiences that help them learn when to use coordinate graphs and when another form of graphing is more appropriate.
Number Pairs
I originally saw this activity done by using a "Number Machine," which is a half-gallon milk carton covered in paper and with two slots in it. An specially prepared index card, with a number on it, is inserted into the top slot of the milk carton and, by going through the carton, the card flips over, to reveal a different number. All of this is to interestingly introduce students to ordered pairs of functional or interrelated numbers. I am sure this concept is better presented in this way in a classroom setting. I just wrote the pairs of numbers on a sheet of paper and Quentin figured them out as logic puzzles. Then he made up a few for me.
Coordinate Tic-Tac-Toe
The next step in learning coordinate graphing is to learn that ordered pairs of numbers can be used to indicate a single point on a grid. James has played the game Battleship many times before, which gave him some experience with ordered pairs of numbers on a grid, but the coordinate points are in the squares and I want him to practice making the coordinate points where the lines intersect.
As I constructed the board for this game, I taught him some terms and the like. I told him that I was numbering the lines that divide the sections and that they are called axes. I showed him that where the two lines crossed, I wrote a zero. On the part of the axis that started at the zero and went to the right, I wrote numbers on each line crossing the axis. I pointed out that the numbers were on the lines and not in the spaces. On the part of the axis that started at the zero and went to the top, I also wrote numbers on the crossing lines. We played Coordinate Tic-Tac-Toe. To win a point, one needs four marks in a
row. To make a move, the player writes down a pair of numbers. The first number tells where the player is starting on the box axis (that's this horizontal line) and going up. The second number tells where the player is starting on the triangle axis (that's this vertical line) and going across. The player then marks (either an X or an O) on the graph paper where the lines for the two numbers meet. It was clear right away that he understood the concepts, but we played for awhile just for the pleasure of it.
Graphing Coordinates
The next task to learn is how to plot sets of numbers from the previous activities on coordinate graph paper and how to use the plotted points to predict future numbers that have the same ordered pattern. I first had him make a coordinate graph on the graph paper just as I had done when we played Coordinate Tic-Tac-Toe. I then gave him 5 or 6 ordered number pairs and had him plot them on the coordinate graph. I then had him draw a line through all points he had marked so far. I then had him predict the additional numbered pairs that could be plotted on the graph, using the same pattern. After doing a few of these, he is now able to make up sets of cards for others to plot.
Now, I have to show him that a set of numbers may yield sets of points that zigzag or curve. The purpose of his learning coordinate graphing is so that he can graph actual events and not all graphable events yield linear patterns. Soon he will begin marking points on graphs to record events not as carefully controlled as the number I select for him. The cards are used to assist the students in learning how to plot points on a coordinate graph, and to show that some activities recorded on a coordinate graph yield points all in a line. The third group of numbers keeps him from thinking this is true for all activities.
Scientific Investigations and Coordinate Graphing
For his first experiences at using his coordinate graphing skills to investigate relationships present in events, I will suggest some scientific investigations.
Rates of change can be measured. As a ball is dropped from higher and higher points, does the height of its bounce change in a predictable manner?
Do all balls bounce to the same height dropped from the same place? Is the graph for each separate ball and its bounce a straight line? Or, do some balls bounce less high as they are moved to successively higher points? What bouncing patterns do things other than balls have?
A study of balls can also lead to the study of circles. Is the diameter of a circle related to its circumference? If the diameter is known, can the circumference be predicted?
We gathered several circular items from around the house. We measured the circumference with a bit of yarn.
Then he measured the diameter of the item.
And we plotted the measurements on a coordinate graph. At some point, he felt confident that there was a relationship between the two measurements and drew a line through the measurements that he had already plotted. This gave him points all along the graph.
After this he could just take one measurement, plot it into the graph and see where the line fell and make a prediction at the second measurement of the pair.
Do squares and rectangles of tiles have predictable perimeters as successive rows or columns of tiles are added?
How would a graph for the perimeter of a rectangle with a constant number of columns and a growing number of rows compare with a graph for the same starting rectangle with growing columns and constant rows?
Perimeter can lead to area. What happens to the area of a rectangle as more rows or columns are added? Does the same thing happen to areas of other shapes? What happens to the perimeter of a rectangle if its area is kept the same but the tiles are rearranged to make long thin rectangles or short fat ones? What numbers would be graphed to show if there was a pattern? If the graph this question produces is not a straight line, can it still be used to make predictions?
Volume and Capacity
Area leads to volume and capacity.
First he measured and cut out different sized cylinders from cardboard tubes. Each cylinder of a different height.
Next he was posed with the question of whether or not a relationship exist between the height of a cylinder and the number of tablespoons or cups of rice it will hold?
Could a coordinate graph be used to predict how many cups of rice a cylinder would hold if only its height were known?
He made a prediction, using the graph that 6 inch cylinder would hold 14 tablespoons, and then confirmed it.
For his own pleasure, he figured out how many tablespoons were in a cup and how this related to each of the measurements. He determined that 1/4 cup equaled 3 tablespoons, so a 1 1/2 inch tube would hold 1/4 cup. He then figured out that since a 2 inch tube holds about 4 1/2 tablespoons and a 4 inch tube holds about 9 tablespoons, then a 6 inch tube would hold about 13 1/2 tablespoons, and he determined that it would be roughly 1 cup.
Capacity can lead to mass or weight. How can a scale give people their weight so quickly? If a rubber band is arranged as shown, can the distance it will stretch as successive numbers of washers are placed on the hook be predicted?
If a graph can predict how far the rubber band will stretch, can it tell how many washers were added?
Other possibilities for extension: Does it make any difference how thick the rubber band is? Is there a limit to how many washers the rubber band will hold? Can the graph be used to tell when this limit is near?
Speed is an area of measurement also. Changes in speed, accelerations or decelerations, lead to graphable relationships.One apparatus for measuring changes in speed can be seen through experimentation.
A toy car is released at the top of the board and allowed to roll freely down the slope. One number indicates the amount of time the car took to make the journey. On the other axis are the inches the board is elevated.
After each successfully timed run at one height, the board is raised one level and the car is timed on a new run. I had them graph the relationship between the time the car takes to get down the slope and the height of the board, and they could then predict the future points on the graph. The same apparatus used to measure acceleration can also be used to measure force. The only additional equipment needed is a block of wood at the end of the ramp. Can a graph be made to predict how far the car will push the block of wood as the car is released from successively higher starting points? If a higher starting point causes the car to knock the wood a predictably farther distance, what would happen if a longer board were used or if the block of wood were heavier or lighter? Do the same things cause the wooden block to be hit farther that also caused the car to travel faster in the experiments on acceleration?
Coordinate graphs are not appropriate in every situation. The students must learn through experience when they are beneficial and when another method of displaying data is more useful. The general rule for distinguishing a topic appropriate to coordinate graphing from topics better represented by another form is to determine if the graph is to be used to make comparisons (other graphs) or to predict
(coordinate graphs). Graphs of any kind are only ways of displaying information; if a student chooses to display data on a coordinate graph and it proves to be of no benefit, he or she may always switch to another form of graphing. Students can learn for themselves when a particular kind of graph is appropriate.
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