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Sovel
[January 30, 1999]
Plane and Simple:
An Introduction to the Cartesian Coordinate System
[including interactive activities]
An article written by
Introduction
Recently, while sharing anecdotes about mathematicians with an associate, she told me a version of the following story: As a young man, Descartes had been resting in bed. He noticed a fly moving on the ceiling. He wondered how he might mathematically be able to describe where the fly was. He realized that he could do it by describing it based on the distance from the walls, utilizing the tiles as units. Thus, this simple problem led to Descartes greatest contribution to mathematics &endash;coordinate geometry1.
As she told me the story, I realized the tiled floor of my classroom could easily be used to demonstrate Descartes' discovery and some of the practical applications of the coordinate plane. As we continued to discuss how we might build on this floor grid analogy, it became evident that this could lead to a series of student discovery activities. The student would not only hear and see strategies that led to some of our current applications, but they also could experience it physically.
The following article is designed to provide a series of activities that can assist the Middle School math teacher in introducing the Cartesian Coordinate system, functions, slope, graphing, scale and proportion. It will provide conceptual and visual activities that reinforce the NCTM Standards, serve as a building block to other subject areas and, perhaps, make it more fun for all involved, at a minimal cost.
Part 1: Using the Plane and Simple floor grid game to apply the NCTM Standards
Part 2: Revisiting René Descartes' Discovery
Part 3: Introduction to the Cartesian Coordinate System
Activity 2b: Speaking mathematically extension...
Activity 3: Hopscotch &endash; preliminary uses of the coordinate plane
Activity 4: Set the table, please...
Activity 5: Intercept them at the pass...
Activity 6: The negative side of plotting...
Activity 7: The name game...
Activity 8: Rise and Run or head'em off at the slope
Activity 9: Let's get functional...
Concluding Remarks Bibliography About the authors
Glossary Constructing the portable axis rolls Return to home
Part 1: Using the Plane and Simple floor grid game to apply the NCTM Standards
Descartes' story can easily be used with Middle School students to illustrate the power of mathematical thinking and modeling. Descartes moved from a simple observation to the desire to express the observation in mathematical terms to then generating a model that demonstrated the observation. This anecdote can lead to a series of increasingly complex and abstract activities that connect and utilize the NCTM standards for middle grades.
The following activities employ the NCTM Standards, which call for experiences with problem solving as a method of inquiry, strategy and application to new situations. The standards encourage communication utilizing a wide range of modalities of expression. They seek applied understanding and spatial reasoning with special attention to proportions and graphs. The middle school student should be able to connect this reasoning and communication to exploring other problems and describing the results. They should be able to represent these relationships numerically and graphically. The middle school student should be able to extend these thoughts by using patterns and functions to represent and solve problems. Therefore, they will be able to connect the expression of these concepts and strategies into the language and processes provided by algebra, statistics, probability and geometry2.
The process of allowing the student to rediscover Descartes' strategy not only leads to the introduction of the rectangular coordinate system, but also to the terms and concepts that go with it. In each case, we can move from a specific to a broader generalization that both connects mathematics to the real world as well as builds a greater foundation for more complex applications.
Part 2: Revisiting René Descartes' Discovery
According to Reshaping School Mathematics: A Philosophy and Framework for Curriculum, "Middle school mathematics should emphasize the practical power of mathematics...Students need to perceive mathematics as more than the subject matter itself &endash; as, in fact, a discipline of reasoning that enables them to attack and solve problems of increasing difficulty and complexity."3
Activity 1: In the beginning...
[vocabulary: Descartes, axis, horizontal, vertical, coordinates, ordered pairs]
Clear an open area in the middle of the classroom. Retell the Descartes anecdote. Explain that today we will use our tiled floor much as Descartes used his ceiling. Show them the "fly" (a beanbag). Toss it out onto the cleared area of your floor. Ask them to write a paragraph on how Descartes might have described the position of the fly.
Discuss their descriptions. In particular, focus on any that use reference points to identify the location. Begin introducing some of the mathematical vocabulary, such as horizontal and vertical, axis, points, coordinates, ordered pairs. List these on the blackboard, with a full, written description. Toss out additional "flies" and have them begin identifying the locations using these terms.
Part 3: Introduction to the Cartesian Coordinate System [ At this point, you should have already completed the construction of the portable axis rolls]
Activity 2a: Speaking mathematically...
[vocabulary: point(s), perpendicular, origin, line]
Now you can lay out your "axis" rolls. Initially lay them out with the axis's forming approximate angles of 45 degrees or 60 degrees. Have your class attempt to isolate the points this way. Next, describe and then show them the ideas of perpendicular and of origin. Have your class attempt to isolate the points this way. Have them write a paragraph discussing the pros and cons of using the perpendicular layout. Have them utilize at least three of the new vocabulary terms. Discuss their conclusions.
Activity 2b: Speaking mathematically extension...
With the grid laid out, review your vocabulary: We see one axis going across horizontally and one going up and down vertically. As you point at each axis, tell them that we will call them "x" and "y", respectively. Where they cross is called the origin or (0,0). The arms of x and y are each divided into separate and equal units, numbered from 1 to infinity [point out some consecutive units]. Note: negative numbers will be discussed in a later activity.
Repeat Activity 1 using the grids. Here is Descartes' fly [show beanbag]. Toss the beanbag out onto the tiled floor. Have your students describe its exact location, in terms of the x and y axis. Repeat several times. For convenience, in the beginning, always have the reference point be the upper right-hand corner of the tile the beanbag lands on. As the student becomes more advanced, have them mark the point where the beanbag lands, moving them away from using nice simple whole numbers to fractions and decimals.
Activity 3: Hopscotch &endash; preliminary uses of the coordinate plane
[vocabulary: height, perimeter, length, unit(s), x-axis, y-axis, plane, Cartesian coordinate system, x-coordinate, y-coordinate]
In activity 2, we saw that each "fly" or point had its own name, expressed as coordinates. This name is always expressed as an ordered pair (x, y). Give them a coordinate. Put the fly [beanbag] at the point that is indicated. List several additional ordered pairs, having a student physically put the beanbag in the appropriate spot(s). List these ordered pairs on a table of values (see table _ example).
You may want to map these points to form a variety of simple geometric shapes, such as a square, rectangle and triangle. If so, have the students use the grid units to describe the measurements of the shapes, at least in terms of length, height and perimeter. You may also utilize this to discuss notions of square units.
By the simple demonstration of tracking "the fly", we have already introduced the concepts of an x-axis (horizontal line) and a y-axis (vertical line) as reference points. Where they cross is at the origin and create a plane called the Cartesian coordinate system. As we isolate the spots that the fly touches, we can express them as point P, made up of an x-coordinate and a y-coordinate. The student has now recreated and used the Cartesian Coordinate System.
Our initial story and discussion communicated in ideas and vocabulary that were basically non-mathematical. We then utilized the mathematical vocabulary for the frames of reference, such as axis, origin, plane, perpendicular, coordinates, or dered pair. Next, we can build on the same simple story to introduce scale and proportion or functions and equations using two variables. Again, we can move from the specific of a particular point (3,5) to the location of any point (x, y) on a plane. It is not a big step, then, to move to idea that two points can be on the same line, that this line can have a name, that the relationship of these coordinates are both predictable and proportional. Indeed, that there exist formulas and procedures to both calculate and duplicate these relationships.
Activity 4: Set the table, please...
[vocabulary: equation, variable, table of values]
As the prior activities demonstrated, the points are simply made up of ordered pairs of x's and y's. These can also be expressed as equations with two variables. For example, let's say the equation is x + y = 4. Then if x = 3 and y = 1, this would fit the equation: 3 + 1 = 4. This also gives us an ordered pair of x and y coordinates. What is it? Place the beanbag on this point.
Are there any other ordered pairs that would fit this equation? What are they? Do they all fit x + y = 4? Have the students design a simple table or chart to help them find out the different values of the variables. Have them find 4 more ordered pairs, using only positive numbers that are > Ø, and then put the beanbags on these points.
Activity 5: Intercept them at the pass...
[vocabulary: y-intercept, x-intercept]
Have the student add these values to their table: if x = Ø, what is the value of y; if y = Ø, what is the value of x. Plot these coordinates on your floor grid. Introduce the idea of "intercepts".
Activity 6: The negative side of plotting...
[vocabulary: negative unit(s)]
Stretch a string or cord along the coordinate points defined in Activity 4 and 5. What shape does it form? Can you find any points from the equation x + y = 4 that are not along this line? What if we made one of the variables a negative number? Does it still fit on the line?
Activity 7: The name game...
If we were then to give this line a mathematical some kind of name, what might a logical name for the line be? Consider that we would want the name to be under stood by any person, even if they spoke a different language. We would also want this name to be able to describe any point on this line.
Would every line have its own unique name? If we had only the line on the grid or graph, what clues would assist us in learning its name?
Activity 8: Rise and Run or head'em off at the slope
[vocabulary: rise, run, slope, ratio, co-efficient]
Refer back to one of the tables generated from Activity 4. Stretch a cord along this line. Using the analogy of a ramp, stretch two additional cords out to form a right triangle base for this ramp or slope. Make sure that the base (or run) is paral lel to the x-axis and the height (or rise) is parallel to the y-axis. Introduce the ratio of a slope, i.e., rise of the ramp/run of the ramp = rise/run. Explain this relation ship as the slope equals the ratio of the rise to the run.
Try this with some other lines and their equations. As they begin to visualize the ratio, draw their attention to any possible clues they can find or see in the equation for the line. Give them an equation for a negative slope. Ask them what the co-efficient of x is in the equation. Have them write a paragraph about any connection that they observe, including about positive and negative slopes.
Activity 9: Let's get functional...
[vocabulary: function, dependent variable, independent variable]
In our simple equations with two variables and with the tables that they have generated, introduce the idea of dependent and independent variables. If we define x as the input or independent variable, then the value of y is always dependent upon what the value of x is.
Now, have them repeat the their table charts by giving them a simple pair of equations, such as y - x = 4 and y = x + 4. Give them several values for x, such as Ø, 1, 2, 3. Have them compute the value of y. Point out that the equations are the same. Have them solve several general equations for y. Remind them that an important property of functions is that for each value of the independent variable, there is only one value of the dependent variable4.
Game 1: Assemble the Puzzle Scavenger Hunt
The teacher will provide several simple jigsaw puzzles. The pieces will be spread out in various places around the room. The teacher will have small boxes or paper bags laid out on various parts of the floor grid. In each container will be a clue to one of the locations of a puzzle piece, as well as the next coordinate point to go to. The teacher will start the Scavenger Hunt by giving each student or team a piece of paper with their particular starting point (coordinates). Each team gains points for finding each piece or item. They gain further points for finding all the pieces. Note: be sure to label the bottom of each container with its unique address, in case it is moved out of place. The student may also look underneath to confirm their coordinate address, but would lose a point each time this is necessary.
Game 2: "What's My Line" Exam
Lay out containers, with questions in them, at various coordinate points on your floor. These points will form a series of lines, some with common intersection points. Each container will have an exam problem. The teacher will give each student an equation. The student will create a table of ordered pairs. They will note the x- and y-intercepts. They will find the line that matches their equation. They will define its slope. They will then go to the points and containers found along their line and answer four of the questions. Intersected points can be extra credit or bonus questions. Thus, the teacher has created several exams for their class while limiting the possibility of students sharing answers throughout the various periods of the day.
Countless activities are available, beyond those shown. The Plane and Simple grid can help visualize the graphing of inequalities as well as the difference be tween positive and negative slopes. The students can see and build models for scale and proportion, as well as exponents and curves. They can graphically learn Pic's Theorem.
Numerous other pos sibilities exist for connecting and applying the Plane and Simple Discovery Activities to student skills and curriculum, particularly with in terdisciplinary teams, such as art and murals, geography and map scales, social sciences and tables/charts, science and func tions/ratios. The activities can be conducted both indoors and outdoors. The suggested activities can be done all at once or two per week over a full unit.
William Glasser points out "...we learn 10% of what we read, 20% of what we hear, 30% of what we see, 50% of what we both hear and see, 70% of what is discussed with others, 80% of what we experience personally and 95% of what we teach someone else." This series of activities not only applies to a wide variety of grade levels, ages and skill levels, but also promotes a multiple intelligence and multiple ability ap proach to learning for the diverse range of students in our modern heterogeneous classroom. Whether the student is RSP, honors, language minority or traditional, these discovery sets all utilize physical movement, written, oral, aural and visual skills and abilities to further their understanding of these concepts. For each of these activities, the students may work individually, in pairs or in teams, as well as indoors or outdoors.
The activities clearly show the NCTM standards come to life. Each activity provides open-ended questions that encourages the full range of student learners to extend their levels of problem solving. By first discussing, then physically do ing each activity, followed by writing about their observations and their conclu sions, we allow the student to own the learning experience. The possibilities of the Plane and Simple Activities not only encourage active involvement and discovery, but lend themselves to the creation of many more activities than could possibly be demonstrated in this article.
(1) Reimer, Luetta and Wilbert. "Historical Connections: Descartes, father of analytic geometry." AIMS, vol. VIII, number 2 (September 1993).: 18-19.
(2) Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: National Council of Teachers of Mathematics, 1989.
(3) Reshaping School Mathematics: A Philosophy and Framework for Curriculum. Washington D.C.: National Academy Press, 1990: 44.
(4) Downing, Douglas. Dictionary of Mathematics Terms. New York: Barron's, 1987.
(5) Basic Algebra. Boston: Houghton Mifflin Company, 1993.
Position: Assistant Professor, Mathematics Department, Sonoma State University, California
Background: BA, MA - Cal State University, Fullerton. Ph.D - Claremont Graduate School (1992).
Teaches all courses in the Statistics-track of the Mathematics major, as well as GE courses in Statistics. Co-director of Statistical Consulting Center at SSU. Also interested in Math Education, especially elementary/middle school curriculum. An active member of the Women In Science Project and the Expanding Your Horizons Conference.
email: susan.herring@sonoma.edu
Position: Mathematics teacher, Petaluma Junior High School, Petaluma, California.
Background: BA, MA - Sonoma State University, California
Teaches seventh grade pre-algebra and eighth grade basic algebra; Petaluma District Mentor teacher in telecommunications; Petaluma District Educational Technology Coordinator.
email: a-teacher@home.com
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[when you have finished with the word you are seeking, click on the word to return to the Activity that you just left or click on your browser's (<-) back key to return to the Glossary.]
axis: the x-axis in the Cartesian coordinates is the line formed when y = 0; the y-axis is the line x = 0.
Cartesian coordinate system: is system where points on a plane are identified by an ordered pair of numbers representing the distances to two perpendicular axes. The horizontal axis is usually called the x-axis,and the vertical axes is usually called the y-axis. The x-coordinate is always listed first in the ordered pair ( x, y ).
coefficient: a term that usually refers to something multiplying something else (usually a constant multiplying a variable).
coordinates: the numbers in an ordered pair representing a point; the coordinates of a point are a set of numbers that identify the location of that point. For example (x = 1, y = 2) are Cartesian coordinates in two-dimensional space
dependent variable: stands for the output number of a function;
Descartes: [1596 - 1650] A French mathematician and philosopher
equation: this is a statement that says two mathematical expressions have the same value. An equation can be put in the general form ax + b = 0, where a and b are known and x is unknown. This is also known as a linear equation, as long as x is always to the power of (+1), not x to the second power [or more] or 1/x.
function: a quantity that depends on a second quantity is a function of it. A function is a rule that turns members of one set into members of another set; an important property of a functon is that for each value of the independent variable there is one and only one value of the dependent variable
height: the distance from the bottom to the top; elevation, above or below a surface, as along the y-axis
horizontal: flat and even, not vertical, as with the x-axis
independent variable: stands for any of the set of input numbers to a function. In the function y = f(x), x is the independent variable and y is the dependent variable
length: the measure of anything from end to end, left to right, as along the x-axis
line: a straight set of points that extnd off into infinity in two directions
negative unit(s): units moving in a direction to the left of the x-axis or downward as measured by the y-axis
ordered pair(s): a set of two numbers; in the Cartesian coordinate system, such a pair would be ( x,y ), where it is agreed that the horizontal coordinate is always listed first and the vertical coordinate is last.
origin: in the Cartesian coordinate system, it is the point where the x - and y- axis intersect; it is the point ( 0,0 )
perimeter: the perimeter is the total distance around a figure, i.e., the sum of all its sides or outer edges.
perpendicular: two lines are perpendicular if the angle between them is a 90 degree angle.
plane: a plane is a flat surface (like a table top) that stretches off to infinity; a plane has zero thickness, but infinite length and width. Any three colliniear points will determine one and only one plane.
point(s): a point on a graph is located by an ordered pair of numbers
ratio: the ratio of any two real numbers a and b is a ÷ b or a/b
rise: the vertical change in moving from point A to point B is the difference between y-coordinates [the change in y]
run: the horizontal change in moving from point A to point B is the difference between x-coordinates [the change in x]
slope: the slope of a line is a number that measures how steep a line is; the ratio of rise to run. A horizontal line has a slope of zero.The slope of a line is defined as the change in y ÷ the change in x [Æy/Æx], where the change in y is the change in the vertical coordinate and the change in x is the change in the horizontal coordinate between any two points on the line. This can also be expressed as y = mx + b, where m = the slope of the line.
table of values: a method to arrange the coordinates of points (ordered pairs) into two columns, one representing the x-coordinates and one representing the y-coordinates; if the ratio of the change in y to the change in x is constant, they are on the same line. See example
unit(s): any fixed quantity, amount, distance, measure, etc. used as a standard; the smallest whloe number on a number line.
variable: a variable is a symbol, usually a letter, that is used to represent a value
vertical: an up-down direction, as in perpendicular to a flat surface; a vertical line has an infinite slope; the y-axis in the Cartesian coordinate system
x-axis: the horizontal axis in a Cartesian coordinate system
x-coordinate: the first coordinate in an ordered pair of numbers; also known as the abscissa.
x-intercept: the value of x at the point where the line or curve crosses the x-axis
y-axis: the vertical axis in a Cartesian coordinate system
y-coordinate: the second coordinate in an ordered pair of numbers; also known as the ordinate.
y-intercept: the value of y at the point where the line or curve crosses the y-axis
Constructing the portable axis rolls
Example of a Table of Values:5
Some equations have two variables. Their solutions are pairs of numbers (ordered pairs), also referred to as a solution pair. If x = 5 and y = 2, then x - y = 3. There are other solution pairs which can fit this equation. They can be both discovered and displayed in a table of values.
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Please send your suggestions, ideas or problems to Barry Sovel