HomeMatching Qualitative Graphs to Story Scenarios Free original lessons
Matching Qualitative Graphs to Story Scenarios
Learn to pair shape-only graphs with narrative descriptions by analyzing rise, fall, and plateau segments. Includes worked example, practice tasks, and a planned card set.
Free lesson
Four exercises
Worked answers
Algebra and Function Practice
Qualitative graphs show the shape of a relationship without numerical scales. Matching a graph to a story means reading each segment as a change: rising, falling, or flat. This lesson teaches a segment-by-segment method, highlights common mismatches, and provides four self-contained practice tasks. A planned cut-out card set with eight story cards and eight graph cards is described in the resource plan; it is not yet built.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
Reading qualitative graphs segment by segment
A qualitative graph uses only shape: the line goes up, down, or stays level. Each segment represents a period in the story. Before matching, divide the graph into segments at every change of direction. Then describe each segment in words: rising means the quantity is increasing, falling means decreasing, and a flat segment means no change.
The horizontal axis usually represents time, but it can represent another ordered quantity. The vertical axis represents the measured quantity, such as distance, speed, temperature, or water level. Because no numbers are shown, you cannot read exact values; you can only compare relative changes. A steeper segment means a faster change than a shallower segment, but steepness alone does not tell you the actual rate.
- Count the segments in the graph and note the order of rise, fall, and plateau.
- Read the story and list the ordered changes it describes.
- Compare the sequence of changes; reject any story whose order differs.
- Check relative steepness only if the story mentions faster or slower change.
- Confirm that the starting and ending behavior match the story's beginning and end.
Using a convention for sketch and card work
When you draw or label qualitative graphs, use this explicit convention: the baseline is the horizontal axis, and the midline is an imaginary horizontal line halfway up the drawing area. A plateau is drawn as a horizontal segment at the current level; it may be at the midline or at any other chosen height above the baseline, as long as it is horizontal. A plateau at the baseline would represent a quantity of zero. A rise is any segment that ends higher than it starts; a fall ends lower. Letter styles and handwriting vary, so accept any clear sketch that follows this convention.
For card matching, write the segment sequence beside each graph, for example rise, plateau, fall. Write the same sequence beside each story. Cards with identical sequences are candidate matches; then use context clues such as faster or slower to decide. Valid alternative matches are allowed when two stories share the same sequence but differ only in unstated details; the answer key should explain why the chosen match is the best fit.
- Draw the baseline and mark the midline lightly.
- Sketch each segment as a straight or gently curved line; label rise, fall, or plateau.
- Write the sequence of segment labels under the sketch.
- For a story card, underline words that indicate increase, decrease, or no change.
- Match by sequence first, then by relative steepness or context.
Checking a match with simple arithmetic
Even without numbers on the graph, you can check consistency using simple arithmetic from the story. If a story says a tank fills for 3 minutes, stays full for 2 minutes, then drains for 4 minutes, the total time is 9 minutes. The graph should have three segments in that order. If the story says the water level rises 2 units per minute for 3 minutes, the rise is 6 units; a fall of 1 unit per minute for 4 minutes is 4 units, so the ending level is 2 units above the start. These hypothetical values are examples only; measure your own equipment if you build physical cards.
Use consistent units throughout. If time is in minutes, keep all time values in minutes. Check that the number of segments equals the number of distinct story phases. Check that a plateau's duration matches the story's stated pause. These checks catch most mismatches before you compare steepness.
- List the story phases with their durations in one unit.
- Count the graph segments and compare with the number of phases.
- Compute any simple totals, such as total time or net change, using the story's numbers.
- Verify that a flat segment appears exactly where the story says nothing changes.
- Record the check result beside the match.
Worked example
A cyclist starts at the bottom of a hill, rides uphill for 5 minutes, rests at the top for 2 minutes, then rides downhill for 4 minutes. Sketch the qualitative graph of distance from the start versus time and describe its segments.
Worked answer
The graph has three segments in order: rise for 5 minutes, plateau for 2 minutes, fall for 4 minutes. If the uphill speed is slower than the downhill speed, the rise is gentler than the fall; if speeds are not stated, any reasonable relative steepness is acceptable under the convention. The plateau is horizontal at the top level. Total time is 11 minutes. Check: three phases match three segments, and the plateau occurs between rise and fall.
Four practice exercises
Exercise 1
Story: A candle is lit and burns steadily for 6 minutes, stays at the same height for 1 minute because the flame is shielded, then burns steadily again for 3 minutes. Describe the segment sequence of the qualitative graph of candle height versus time.
Show answer and check
Fall, plateau, fall.
Why: Burning steadily decreases height, so the first segment falls. Shielding stops the change, giving a plateau. Burning again decreases height, giving a second fall. The order is fall, plateau, fall.
Check: Count story phases: burn, pause, burn = 3 phases. Count segments in the answer: fall, plateau, fall = 3 segments in matching order. Total time 6 + 1 + 3 = 10 minutes is consistent with three time intervals.
Exercise 2
Story: A runner jogs at a steady speed for 4 minutes, sprints faster for 2 minutes, then walks slowly for 5 minutes. The graph shows speed versus time. Describe the segment sequence and relative steepness.
Show answer and check
Plateau, rise, plateau, fall, plateau. The first plateau is steady jogging speed; the rise shows speeding up to sprint; the second plateau is steady sprinting speed; the fall shows slowing to walking; the final plateau is steady walking speed. Relative steepness of the rise and fall is not specified.
Why: Jogging at a steady speed means speed is constant, so the first segment is a plateau. Speeding up to sprint increases speed, so the next segment rises. Sprinting at a steady faster speed is another constant speed, so it is a second plateau at a higher level. Slowing to walk decreases speed, so the next segment falls. Walking at a steady slow speed is a final plateau at a lower level. The story does not state how quickly the runner changes speed, so no relative steepness should be required. The sequence is plateau, rise, plateau, fall, plateau.
Check: List phases: steady jog, speed up to sprint, steady sprint, slow to walk, steady walk = 5 phases. The answer has 5 segments in matching order: plateau, rise, plateau, fall, plateau. Check that the first and second plateaus are flat, with the second higher than the first, and that the final plateau is flat and lower than the sprint plateau. Do not require a specific steepness for the rise or fall, since the story does not state how quickly the runner changes speed. Total time 4 + 2 + 5 = 11 minutes matches the three stated durations across five phases.
Exercise 3
Story: A water tank starts half full. It fills for 3 minutes, stays full for 2 minutes, then drains for 4 minutes until empty. The graph shows water level versus time. Describe the segment sequence and the starting level.
Show answer and check
Rise, plateau, fall; the graph starts at a positive level, not at zero.
Why: Filling increases the level, so the first segment rises from a positive starting height. Staying full gives a plateau at the top. Draining decreases the level to zero, so the final segment falls to the baseline. The starting level is positive because the tank is half full.
Check: Three phases give three segments. The plateau is between rise and fall. The final level is zero, so the last segment ends at the baseline. Total time 3 + 2 + 4 = 9 minutes matches three intervals.
Exercise 4
Story: A hiker climbs for 2 minutes, rests for 3 minutes, climbs faster for 1 minute, then descends for 4 minutes. The graph shows elevation versus time. Describe the segment sequence and identify which rise is steeper.
Show answer and check
Rise, plateau, steeper rise, fall. The third segment is the steeper rise.
Why: Climbing increases elevation, so the first segment rises. Resting gives a plateau. Climbing faster gives a second rise that is steeper than the first. Descending decreases elevation, so the last segment falls. The sequence is rise, plateau, steeper rise, fall.
Check: Four phases give four segments. The plateau is second, and the steeper rise is third. Total time 2 + 3 + 1 + 4 = 10 minutes matches four intervals. Check that the final segment falls and ends lower than the highest point.
Common mistakes
- Matching by overall direction only, ignoring the order of segments. Write the full sequence of rise, fall, and plateau for both graph and story before deciding.
- Treating a steep rise as always meaning a large total increase. Steepness shows a faster change, not a larger total; check the segment's duration in the story.
- Assuming every flat segment means the quantity is zero. A plateau means no change at the current level, which may be high, low, or zero.
- Ignoring the starting level described in the story. Check whether the story begins at zero, at a positive level, or already changing.
Self-check rubric
- Graph is divided into segments at every change of direction.
- Each segment is labeled rise, fall, or plateau.
- The sequence of segments matches the order of events in the story.
- Relative steepness is used only when the story gives comparative speed or rate information.
- Starting and ending behavior are consistent with the story.
- Sketch follows the baseline-midline convention and is clear enough to read.
- Valid alternative sketches with the same segment sequence are accepted when the story does not specify relative steepness.
Questions and answers
Can two different stories match the same qualitative graph?
Yes, if they share the same sequence of rise, fall, and plateau. The best match uses additional context such as relative steepness or stated durations. The answer key should explain the reasoning rather than declare a single universal answer.
Do I need numbers on the axes to match a graph to a story?
No. Qualitative matching uses only shape and order of segments. Numbers would make it quantitative, which is outside this task.
How do I show a pause in a story on a qualitative graph?
Draw a horizontal segment, called a plateau, at the current level. Its length can suggest the pause duration, but without a scale the length is only relative.
What if the story says the quantity rises faster later?
Draw the later rising segment steeper than the earlier one. Steepness is comparative only; no numerical slope is required.