084i - Describing Relationships in Graphs

Lesson 84 — Describing Relationships Between Variables in Graphs

Strand: Algebra | Descriptor: AC9M7A04 | Duration: 45 minutes

Learning Intentions

• To describe how one variable changes with another, using precise language.
• To match graphs to the situations they represent, and justify the match.

Success Criteria

I can:

1. Identify the independent and dependent variables in a situation.
2. Describe a relationship using increasing/decreasing, steep/gradual, constant.
3. Match a graph to a story and justify each feature.
4. Sketch a graph from a verbal description.

Warmup

(6 minutes — which variable depends? pairs)

For each pair, decide which variable depends on the other, and which axis each belongs on.

1. Time walking; distance covered.
2. Number of guests; cost of catering.
3. Age of a car; its resale value.
4. Hours of revision; test score.

Answers: dependent variables — distance, cost, value, score (these go on the vertical axis). The independent variable (time, guests, age, hours) goes on the horizontal.

The convention to name: the horizontal axis carries the variable we choose or that simply passes (time); the vertical carries what responds.

Activities

Activity 1 — Explicit Instruction: the Language of Relationships (12 min)

Build the description toolkit on one worked example — a car journey graph (speed vs time): accelerates from rest, cruises steadily, slows for a town, cruises slower, brakes to a stop.

Sentence frames to teach:

• “As [independent] increases, [dependent] increases/decreases/stays constant.”
• “The graph is steepest when …, meaning … is changing fastest.”
• “The flat section shows …”

I do — describe the car graph in exactly four sentences, one per phase, using the frames. Then condense to one summary sentence: “Speed rises quickly, holds steady, dips through the town, and falls to zero at the end.”

Distinguish two levels of description:

Level	Example	When to use
Feature-by-feature	”From – s the speed increases steeply…”	Detailed analysis
Overall trend	”Speed generally decreases across the trip.”	Summaries and comparisons

We do — the cooling cuppa. A graph shows tea temperature falling fast at first, then ever more slowly towards room temperature (C).

1. Independent and dependent variables?
2. Describe the relationship in one sentence.
3. Why does the graph flatten but never quite reach C — what does that say about the situation?

(Time; temperature. “As time passes, temperature falls — quickly at first, then more and more slowly.” The tea approaches room temperature; once close, there is little difference left to drive cooling.)

Activity 2 — Graph–story Matching (14 min)

Pairs. Six graphs, six stories — but one story has no matching graph and one graph has no story, forcing justification rather than elimination.

The stories:

1. Water depth in a bath: filled steadily, soak, then drained.
2. Distance from home on a walk to the shop and straight back.
3. Temperature of an oven preheating, holding, then switched off.
4. Height of a flag being raised in short pulls with pauses.
5. Number of spectators in a stadium before, during and after a match.
6. Depth of water in a leaking bucket being filled faster than it leaks.

The graphs (described for the teacher; draw on cards):

A — rises steadily, flat, falls steeply. B — rises, flat top, falls to zero, symmetric tent shape. C — staircase: rise, flat, rise, flat. D — rises quickly then flattens at a high level, later decays. E — steady rise throughout, no pauses. F — falls steeply then levels near zero.

Matches: 1–A; 2–B; 3–D; 4–C; 5 — no graph (would need rise, long flat, fall — close to A but the fall should not start from the fill level’s timing; the honest answer is “A is the nearest but its proportions fit the bath better; no exact match”); 6–E. Graph F matches no story (it fits e.g. the cooling cuppa).

Teacher note: the deliberate near-miss between stories 1 and 5 is the point — both are rise/flat/fall. The justification (what each phase’s steepness and duration should look like) is what earns the match, not the general shape. Accept 5–A with strong justification and 1 left unmatched: the argument is the assessment.

Socratic prompts while circulating:

Prompt	Purpose
Which feature of the graph is the strongest clue?	Trains feature-first matching, not vibes.
Could two stories share this graph? What would distinguish them?	Surfaces the 1 vs 5 clash productively.
For the leaking bucket — why is the rise steady rather than curved?	Fill rate minus leak rate is roughly constant.
What story would fit the leftover graph?	Reverses the skill: graph to story.

Activity 3 — Sketch from a Story (8 min)

Individually. Axes labelled, no numbers required — shape is the content.

Sketch a graph for each:

1. The height of grass over a month, mown flat every Saturday.
2. Your distance from the classroom door during a fire drill (walk out, wait at assembly, walk back).
3. The value of a phone from new over three years.

Expected shapes: 1 — sawtooth: steady rises cut by sharp drops; 2 — rise, long flat, fall back to zero; 3 — falling curve, steep early, flattening later.

Gallery check: swap sketches; the reader writes the story they see, and the pair compares against the intended one. Mismatches locate ambiguous features.

Checks for Understanding

(5 minutes — exit ticket)

1. In “hours worked vs pay”, which is the dependent variable, and which axis does it take?
2. Describe in one sentence: a graph that rises steeply, then rises gradually, then is constant.
3. Sketch: the temperature of a glass of iced water left in a warm room.
4. A story says “she ran, rested, then walked home.” What three features must its distance-from-home graph show?
5. Reasoning. Two different stories can match the same general graph shape. What breaks the tie?

Answers: 1. Pay; vertical; 2. E.g. “The quantity grows quickly at first, then more slowly, then stops changing”; 3. Rising curve, steep then flattening toward room temperature; 4. Steep rise, flat section, gentler fall back to zero; 5. The proportions and rates — how steep and how long each phase should be, argued from the situation.

Common Misconceptions

Misconception	How to pre-empt it
Reading the graph as a literal picture (a hill in the graph = a hill on the walk).	Named in L83; here the walk story (2) tests it — distance-from-home rises and falls with no hill involved.
Swapping the axes.	The warmup’s dependence sort; enforce the convention every sketch.
Believing steeper always means “more”.	Steeper means faster change; the flat section of D is the highest but changes least.
Vague descriptions (“it goes up then down a bit”).	Sentence frames make precision the default.
Matching by overall shape alone.	The deliberate 1 vs 5 clash requires feature-level justification.
Sketching with unnecessary wiggles.	Shape communicates claims; every bend must mean something.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A tap fills a bottle with straight sides, then a narrower neck. Sketch depth vs time and explain the change in steepness.

Answer

Steady rise, then a steeper steady rise: the same inflow raises the level faster in the narrow neck.

E2 (AMC Junior style). Which fills with depth rising at a decreasing rate under constant inflow: a cylinder, a cone point-down, or a cone point-up?

Answer

Cone point-up: the container widens as it fills, so each centimetre of depth needs more water. (Point-down gives an increasing rate; the cylinder is constant.)

E3 (Challenge). Runner A’s distance–time graph is a straight line; Runner B’s curves upward, starting flatter and ending steeper, finishing at the same point at the same time. Who led at halfway (by time), and do the graphs ever cross before the finish?

Answer

A led at halfway — B’s flatter start means less distance covered early. The graphs meet only at the start and the finish: B’s curve lies below A’s line the whole way between (slower early, faster late, equal totals), so B never leads and there is no crossing before the line. B draws level exactly at the finish — the closest possible race that is never actually tied mid-race.

E4 (Challenge). Sketch speed vs time for a ball thrown straight up and caught. Then sketch height vs time. Why do the two graphs peak at different moments?

Answer

Speed: starts high, falls to zero (at the top), rises again (drawn as size of speed). Height: tent-curve peaking in the middle. Height peaks exactly when speed is zero — the peaks cannot coincide because one variable’s maximum is the other’s minimum.

Homework

1. For each situation, name the independent and dependent variables: (a) minutes on a treadmill vs kilojoules burned (b) number of toppings vs pizza price (c) days since planting vs seedling height.
2. Describe each graph in one or two sentences using the lesson’s vocabulary: (a) rises steadily throughout (b) falls steeply, then levels out (c) alternates flat and rising sections.
3. Sketch, with labelled axes: (a) depth of water in a sink filled, used for washing up (level falls a little with splashing), then drained (b) the number of people at a bus stop across an hour with buses every minutes.
4. Match each story to a shape and justify in one sentence: a candle burning down; savings with weekly deposits; a population of bacteria doubling regularly.
5. Reasoning. A graph of “distance from home” for a jogger goes up, is flat, then goes up more steeply. A student says the flat part means the jogger walked slowly. Correct this.
6. Reasoning. Explain the difference between a graph being high and being steep, with an example where the highest section is the least steep.
7. Challenge. Sketch depth vs time for a constant-rate fill of a glass that is a hemisphere on a stem (narrow stem, then widening bowl). Annotate where and why the steepness changes.

Answers: Q1 — dependent: kilojoules, price, height. Q4 — candle: steady fall; savings: staircase; bacteria: increasingly steep rise. Q5 — flat means distance unchanged: the jogger was stationary; walking slowly would still rise, just gently. Q6 — high = large value; steep = fast change; e.g. a car at cruising speed: the speed graph is high and flat. Q7 — very steep in the stem (little water per cm), progressively gentler as the bowl widens, steepening again past the bowl’s widest point.