An X-Y coordinate plane manipulative provides a hands-on way to plot and manipulate points, lines, and polygons in the x-y coordinate plane. A Coordinate planes manipulative provides a concrete and visual way for students to understand ordered pairs, quadrants, integer signs, distances, absolute value, ratios, proportions, area, linear equations, and line graphs.
Physical coordinate planes typically use a pegboard where pegs model points and rubber bands model line segments and graph lines. These X-Y Coordinate Pegboards include sliding x- and y-axes that can be positioned to make 1- or 4-quadrant coordinate planes of different sizes.
Virtual manipulatives, such as Brainingcamp's Coordinate Plane Manipulative, offer a much greater area and range of values (600 points compared to 225 points on a typical physical x-y pegboard). Brainingcamp's virtual coordinate plane also includes labeled axes and point coordinates, as well as custom graphing modes.
Graphing Ordered Pairs
A coordinate plane is formed when two number lines intersect at their zero points. The horizontal line is the x-axis and the vertical number line is the y-axis. An ordered pair describes a point on the coordinate plane. The x-coordinate (first number) tells how far to move along the x-axis. The y-coordinate (second number) tells how far to move along the y-axis. To graph the point (7,4), start by placing a point at the origin (0,0). Then move the point 7 units right and 4 units up.
Quadrants and Integer Signs
The axes divide the plane into four regions called quadrants. The sign of the x- and y-coordinates identify the quadrant in which a point is located. When the x- or y-coordinates of two points have opposite signs, the points are as reflected across that axis. The points are the same distance, in opposite directions, from the axis across which they are reflected.
Distance Between Points
When two points share an x- or y-coordinate, they form a straight line. To find the distance between points that share an x- or y-coordinate, count the number of units between them. The distance between points that do not share an x- or y-coordinate can be found using the Distance Formula, which comes from the Pythagorean Theorem. The distance d between points (x1, y1) and (x2, y2) is d = √((x2 – x1)² + (y2 – y1)²).
The distance between points that share an x- or y-coordinate can be found by counting the number of units between them. To compute this distance, find the absolute value of the difference in their uncommon difference. The distance between points (-4, 3) and (6, 3) is |-4 – 6|, or 10.
Ratios and Proportional Relationships
Two quantities have a proportional relationship if their ratio is constant. Quantities are proportional if the graph of their values is a straight line that goes through the origin. The constant ratio is called the “constant of proportionality”, the “slope” of the graphed line, or the “unit rate”. The unit rate is the value r of the point (1, r) along the graphed line. Plot points to see if they have a proportional relationship.
Area is the amount of space that a flat surface occupies. Area is measured by how many unit squares fit inside a figure. Each small square on the coordinate plane is a unit square because it is one unit long and one unit high. The area of figures on the coordinate plane can be found or estimated by counting the number of unit squares inside the figure.
Graphing Linear Equations
A linear equation is an equation whose graph is a straight line. Linear equations can be written in the form y = mx + b. The variable b is called the y-intercept and is the point where the line crosses the y-axis. The variable m is called the slope and is the ratio of the change in the y-values to change in the x-values between any two points on the line.