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Graph a Line from a Point and Slope: Rise Over Run Practice
Learn to plot a line starting at a given point using slope as rise over run, with clear steps, common fixes, and four self-contained practice tasks.
Free lesson
Four exercises
Worked answers
Algebra and Function Practice
Graphing a line from one point and a slope is a direct application of rise over run. You begin at the specified point, use the slope to find a second point, and connect them with a straight line. This lesson defines a consistent plotting convention, shows a worked example, and provides four practice tasks with reasoning and check methods. All numerical values are simple and units are consistent. The planned worksheet and answer key are described but not yet built.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
Understanding Point and Slope
A slope is a ratio of vertical change (rise) to horizontal change (run). A slope of 3/4 means rise 3 units for every run 4 units. A negative slope means the line falls as it moves right. The given point is the anchor: every plotted line must pass through it.
Use a coordinate grid with equal spacing on both axes. Label the axes and mark the given point clearly. Choose a scale that fits the rise and run without running off the grid. For example, if the run is 5 and the grid has 10 squares across, a scale of 1 unit per square works.
- Plot the given point and label it with its coordinates.
- Write the slope as a fraction rise/run, keeping the sign with the rise.
- From the given point, move vertically by the rise and horizontally by the run to mark a second point.
- Repeat the move in the same direction to mark a third point for accuracy.
- Draw a straight line through all marked points and extend it across the grid.
Plotting Convention and Valid Alternatives
Convention used here: each grid square represents one unit unless a scale is stated. The x-axis is horizontal, the y-axis is vertical, and the origin is at the intersection. Points are marked with a small dot and labeled with coordinates. Lines are drawn with a straightedge.
Valid alternatives: you may move the run first and then the rise, as long as the net change matches the slope. You may also use an equivalent fraction, such as 6/8 for 3/4, if the grid allows. Handwriting and dot size vary; the rubric checks correct coordinates and a straight line through them, not a single universal appearance.
- State the scale if it is not one unit per square.
- Mark at least two points that satisfy the slope from the given point.
- Check that the line passes through the given point and the second point.
- Verify the slope by counting rise and run between any two marked points.
Checking Your Graph
A graph is correct if the line passes through the given point and the rise/run between any two marked points equals the slope. Count squares carefully and confirm the sign: a negative slope should fall from left to right.
Simple arithmetic check: if the given point is (2, 1) and the slope is 3/2, then moving run 2 and rise 3 gives (4, 4). The change in y divided by the change in x is (4 - 1)/(4 - 2) = 3/2, which matches.
- Re-count the rise and run between two marked points.
- Substitute the coordinates into rise/run and simplify.
- Confirm the line extends in both directions and is straight.
Worked example
Graph the line through (1, 2) with slope 3/2. Use one unit per grid square.
Worked answer
Plot (1, 2). Move rise 3 and run 2 to reach (3, 5). Move again to reach (5, 8). Draw a straight line through (1, 2), (3, 5), and (5, 8). Check: (5 - 2)/(3 - 1) = 3/2.
Four practice exercises
Exercise 1
Graph the line through (2, 1) with slope 1/2. Use one unit per grid square. Describe the points you plot.
Show answer and check
Plot (2, 1). Move rise 1 and run 2 to reach (4, 2). Move again to reach (6, 3). Draw a straight line through these points.
Why: Slope 1/2 means rise 1 for run 2. Starting at (2, 1), adding run 2 and rise 1 gives (4, 2), then (6, 3). The line rises gently from left to right.
Check: Count between (2, 1) and (4, 2): rise 1, run 2, so slope 1/2. Confirm the line passes through both points and is straight.
Exercise 2
Graph the line through (0, 4) with slope -2/3. Use one unit per grid square. Describe the points you plot.
Show answer and check
Plot (0, 4). Move rise -2 and run 3 to reach (3, 2). Move again to reach (6, 0). Draw a straight line through these points.
Why: Slope -2/3 means fall 2 for run 3. From (0, 4), moving right 3 and down 2 gives (3, 2), then (6, 0). The line falls from left to right.
Check: Between (0, 4) and (3, 2): change in y is -2, change in x is 3, so slope -2/3. Confirm the line passes through both points.
Exercise 3
Graph the line through (3, 3) with slope 2. Use one unit per grid square. Describe the points you plot.
Show answer and check
Plot (3, 3). Write slope 2 as 2/1. Move rise 2 and run 1 to reach (4, 5). Move again to reach (5, 7). Draw a straight line through these points.
Why: A whole-number slope is written over 1. Rise 2 and run 1 from (3, 3) gives (4, 5), then (5, 7). The line is steep and rises from left to right.
Check: Between (3, 3) and (4, 5): rise 2, run 1, so slope 2. Confirm the line passes through both points and extends straight.
Exercise 4
Graph the line through (4, 2) with slope -1/4. Use one unit per grid square. Describe the points you plot.
Show answer and check
Plot (4, 2). Move rise -1 and run 4 to reach (8, 1). From (4, 2), you can also move left 4 and up 1 to reach (0, 3). Draw a straight line through (0, 3), (4, 2), and (8, 1), extending it across the grid.
Why: Slope -1/4 means fall 1 for run 4. From (4, 2), moving right 4 and down 1 gives (8, 1); moving left 4 and up 1 gives (0, 3). Both moves match the slope, and all three points lie within a 10 by 10 grid at one unit per square. The line falls gently from left to right.
Check: Between (4, 2) and (8, 1): change in y is -1, change in x is 4, so slope -1/4. Between (0, 3) and (4, 2): change in y is -1, change in x is 4, so the slope is again -1/4. Confirm the line passes through all three points and is straight.
Common mistakes
- Swapping rise and run, for example moving 4 up and 3 right for slope 3/4. Write the slope as rise/run before moving. The numerator is vertical, the denominator is horizontal.
- Dropping the negative sign and drawing a line that rises instead of falls. Keep the sign with the rise. For -2/3, move down 2 and right 3, or up 2 and left 3.
- Starting from the origin instead of the given point. Mark and label the given point first, then move from that point.
- Using unequal spacing on the axes without stating a scale. State the scale and keep square spacing equal on both axes.
Self-check rubric
- The given point is plotted and labeled correctly.
- At least one additional point is plotted using the correct rise and run.
- The sign of the slope is applied correctly.
- A straight line passes through all plotted points and extends across the grid.
- The scale is stated if it is not one unit per square.
- Valid alternative point orders or equivalent fractions are accepted if the slope and line are correct.
Questions and answers
Can I move left instead of right?
Yes. Moving left by the run and down by the rise gives the same slope, because both changes are reversed. The line is the same.
What if the slope is a whole number like 2?
Write it as 2/1. Rise 2, run 1. The line is steeper than a slope of 1/2.
How many points do I need?
Two points determine a line, but marking a third helps catch counting errors. The rubric accepts two correct points and a straight line.
Does the line have to start at the given point?
The line must pass through the given point, but it can extend beyond it in both directions. The given point is an anchor, not an endpoint.