Plot Two Points, Draw the Line, and Compute Slope

Algebra and Function Practice

This lesson teaches a reliable procedure for graphing a line from two ordered pairs and finding its slope. You will plot each point, draw the line through them, mark the vertical change (rise) and horizontal change (run), and compute slope as rise divided by run. The method works for any two distinct points, including points with negative coordinates. A consistent convention is used throughout: subtract coordinates in the same point order in both numerator and denominator, and use one uniform grid scale per graph. All numerical examples are hypothetical and chosen for easy arithmetic; measure your own grid and tools when applying the method.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

Plotting Ordered Pairs on a Grid

An ordered pair (x, y) gives a horizontal position x and a vertical position y. Start at the origin (0,0). Move along the x-axis by the first number: right for positive, left for negative. Then move parallel to the y-axis by the second number: up for positive, down for negative. Mark the point where these two moves meet.

Use a sharp pencil and a straightedge. Label each point with its coordinates so you can check your work. If a coordinate is zero, the point lies on an axis: (0, y) is on the y-axis, and (x, 0) is on the x-axis.

1. Draw and label the x-axis and y-axis; mark the origin.
2. For the first ordered pair, move horizontally from the origin by the x-value.
3. From that position, move vertically by the y-value and mark the point.
4. Label the point with its ordered pair.
5. Repeat for the second ordered pair.

Drawing the Line and Marking Rise and Run

Place a straightedge so it passes through both plotted points. Draw a straight line through them and extend it a little beyond each point. The line represents all points that satisfy the same linear relationship.

To prepare for slope, choose one point as the start and the other as the end. Draw a vertical segment from the start point to the horizontal level of the end point; this is the rise. Draw a horizontal segment from that level to the end point; this is the run. Label the rise and run with their signed values or with arrows showing direction.

1. Align a straightedge through both points and draw the line.
2. Mark the start point and the end point on the line.
3. Draw a vertical segment for the rise from the start point to the end point's horizontal level.
4. Draw a horizontal segment for the run from the end of the rise to the end point.
5. Label the rise and run, noting whether each is positive (up or right) or negative (down or left).

Computing Slope as Rise over Run

Slope measures steepness and direction. Using two points (x1, y1) and (x2, y2), slope m = (y2 - y1) / (x2 - x1). The numerator is the vertical change (rise), and the denominator is the horizontal change (run). Keep the order of subtraction consistent: if you subtract the first point's coordinates from the second, do the same for both x and y.

A positive slope rises from left to right; a negative slope falls from left to right. A slope of zero is horizontal, and an undefined slope occurs when the run is zero (a vertical line). Simplify the fraction when possible, and check that the sign matches the direction you marked on the grid.

1. Identify (x1, y1) and (x2, y2) from the two plotted points.
2. Compute the rise: y2 - y1.
3. Compute the run: x2 - x1.
4. Divide rise by run to get slope m.
5. Simplify the fraction and compare its sign with the drawn line.

Worked example

Plot the points (1, 2) and (4, 8), draw the line through them, mark rise and run, and compute the slope.

Four practice exercises

Common mistakes

• Subtracting coordinates in different orders, such as y2 - y1 but x1 - x2. Keep the same point order in both numerator and denominator. Write the formula with labels before substituting.
• Plotting (x, y) as (y, x) by moving vertically first. Always move horizontally by the x-value first, then vertically by the y-value. Say the pair aloud as 'x, then y'.
• Ignoring negative signs when counting rise or run. Mark direction arrows on the grid and record signed values. A leftward run is negative; a downward rise is negative.
• Dividing run by rise instead of rise by run. Remember slope is rise over run: vertical change divided by horizontal change. Write 'rise/run' above the calculation.

Self-check rubric

• Both points are plotted at the correct coordinates and labeled.
• A straight line passes through both points and is drawn with a straightedge.
• Rise and run are marked and labeled with correct direction or signed values.
• Slope is computed as rise divided by run and simplified correctly.
• The sign of the slope matches the direction of the drawn line.
• Conventions are consistent: the same point order is used in both numerator and denominator, and one uniform grid scale is used per graph.

Questions and answers