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Learn to draw lines on graph paper from a given slope and starting point, mark rise and run, and label the slope correctly.
This lesson teaches you to draw a straight line on a coordinate grid when you are given a starting point and a slope value. You will mark the rise and run directly on the graph paper, then label the slope as a fraction. The convention used here is that slope equals rise over run, where rise is the vertical change and run is the horizontal change. A positive slope rises from left to right; a negative slope falls from left to right. You will practice with simple integer and fractional slopes, check your work by counting grid squares, and use a consistent labeling method. All examples use a standard square grid where each small square represents one unit. No special tools are required beyond graph paper and a pencil.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
Slope measures steepness. On a coordinate grid, slope is the ratio of vertical change (rise) to horizontal change (run). If a line goes up 2 units for every 3 units to the right, its slope is 2/3. If it goes down 1 unit for every 4 units to the right, its slope is -1/4. A horizontal line has slope 0; a vertical line has undefined slope, which is outside this lesson.
When you draw a line from a starting point, you use the slope to find a second point. From the starting point, move vertically by the rise and horizontally by the run, then mark the new point. Connect the two points with a straight line. Always move left-to-right for the run to keep the convention consistent.
Suppose the starting point is (1, 1) and the slope is 2/3. The rise is 2 and the run is 3. From (1, 1), move up 2 units to y = 3, then right 3 units to x = 4. The second point is (4, 3). Draw a line through (1, 1) and (4, 3). To check, pick another point on the line, such as (7, 5), and confirm the rise from (4, 3) to (7, 5) is 2 and the run is 3.
Label the line with the fraction 2/3 and optionally mark a small right triangle showing the rise and run between two grid points. The triangle helps others see the slope without recalculating.
A negative slope means the line falls from left to right. For slope -1/2 with starting point (0, 4), the rise is -1 and the run is 2. From (0, 4), move down 1 unit to y = 3, then right 2 units to x = 2. The second point is (2, 3). Draw the line and label it -1/2. You can also write the negative sign in the numerator, as in -1/2, or in front of the fraction, as in -(1/2). Both are acceptable if the label clearly shows a negative slope.
To avoid confusion, always move right for the run. If you move left, you must also reverse the rise sign, which changes the direction and can lead to errors. Stick to rightward runs.
Label the slope close to the line, using a fraction with rise on top and run on the bottom. For integer slopes, write the fraction with denominator 1, such as 3/1, or write the integer with a note that run is 1. Keep labels legible and avoid crossing grid lines. A small rise-run triangle drawn between two grid points on the line is a valid alternative labeling method; the triangle should show the vertical and horizontal legs with lengths matching the rise and run.
Check your work by selecting two points on your drawn line that are exactly on grid intersections. Count the vertical change and horizontal change between them. The ratio should simplify to the given slope. If it does not, recheck your point placement and redraw.
Starting point (2, 1), slope 3/4. Draw the line on graph paper, mark rise and run, and label the slope.
From (2, 1), move up 3 units to y = 4, then right 4 units to x = 6. Mark (6, 4). Draw a straight line through (2, 1) and (6, 4). Label the slope as 3/4. Check: from (2, 1) to (6, 4), rise = 3 and run = 4, so slope = 3/4.
Starting point (0, 0), slope 1/2. Draw the line, mark rise and run, and label the slope.
From (0, 0), move up 1 and right 2 to (2, 1). Draw the line through (0, 0) and (2, 1). Label 1/2.
Why: Slope 1/2 means rise 1, run 2. Moving up 1 and right 2 from the origin gives (2, 1). A straight line through these points has the required slope.
Check: Pick (4, 2) on the line. Rise from (2, 1) to (4, 2) is 1, run is 2, ratio 1/2. Correct.
Starting point (1, 3), slope -2/3. Draw the line, mark rise and run, and label the slope.
From (1, 3), move down 2 and right 3 to (4, 1). Draw the line through (1, 3) and (4, 1). Label -2/3.
Why: Negative slope means fall from left to right. Rise is -2, run is 3. From (1, 3), down 2 gives y = 1, right 3 gives x = 4.
Check: From (1, 3) to (4, 1), rise = -2, run = 3, slope = -2/3. The line falls as x increases.
Starting point (2, 2), slope 3. Draw the line, mark rise and run, and label the slope.
Write slope as 3/1. From (2, 2), move up 3 and right 1 to (3, 5). Draw the line through (2, 2) and (3, 5). Label 3/1 or 3.
Why: Integer slope 3 equals 3/1, so rise 3, run 1. Moving up 3 and right 1 from (2, 2) gives (3, 5).
Check: From (2, 2) to (3, 5), rise = 3, run = 1, slope = 3. Another point (4, 8) also fits.
Starting point (0, 5), slope -1/4. Draw the line, mark rise and run, and label the slope.
From (0, 5), move down 1 and right 4 to (4, 4). Draw the line through (0, 5) and (4, 4). Label -1/4.
Why: Slope -1/4 means rise -1, run 4. From (0, 5), down 1 gives y = 4, right 4 gives x = 4.
Check: From (0, 5) to (4, 4), rise = -1, run = 4, slope = -1/4. The line falls gently from left to right.
Write it as 4/1. The rise is 4 and the run is 1. From the starting point, move up 4 and right 1.
Fractions are preferred because they show rise and run directly. A decimal like 0.5 is acceptable only if you also show the fraction 1/2, since the drawing convention relies on counting squares.
Two points determine a straight line. Use the starting point and one point found from the slope. You may mark a third point as a check.
In this lesson, all starting points are on grid intersections. If one is not, move to the nearest grid intersection and adjust, but that is outside the current scope.