085i - Interpreting Graphs in Context

Lesson 85 — Interpreting Graphs in Real-World Contexts

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

Learning Intentions

• To draw conclusions from graphs and justify them with graph features.
• To compare two data sets plotted on the same axes.

Success Criteria

I can:

1. Answer contextual questions using evidence from a graph.
2. Compare two lines on shared axes and interpret their intersection.
3. Calculate simple rates from graph readings.
4. Distinguish what a graph shows from what it merely suggests.

Warmup

(6 minutes — claim check, projected)

A graph shows two mobile plans’ monthly cost vs data used: Plan A starts at 20$08$ GB.

For each claim: supported, contradicted, or can’t tell from the graph?

1. “Plan B is cheaper for light users.”
2. “Plan A has better network coverage.”
3. “At GB the plans cost the same.”
4. “Plan B is always the worse choice.”

Answers: 1. Supported — B’s line is lower left of the crossing; 2. Can’t tell — coverage is not on the graph; 3. Supported — that is what crossing means; 4. Contradicted — B is cheaper below GB.

The discipline to name: every claim must point at a graph feature. Claims about things not plotted are outside the graph’s authority.

Activities

Activity 1 — Explicit Instruction: Intersections and Comparisons (12 min)

Two lines on shared axes — the three questions worth asking:

1. Where is each line higher? (Who leads, when?)
2. Where do they cross? (The values are equal — the break-even or overtaking point.)
3. Which is steeper? (Whose value changes faster?)

I do — the phone plans, quantified. Plan A: 20$2$4.50$/GB.

Read from the graph, then verify by calculation:

Connect explicitly to Lessons 25–26: the graphical crossing is the equation’s solution. The graph shows it; algebra confirms it. Two representations, one fact.

Rates from graphs — the second skill. From a distance–time graph, speed distance change time change over a straight section:

We do: From a printed graph of two swimmers’ distance vs time in a m race (A steady throughout; B fast start, fades):

1. Who led at s?
2. When did A overtake B?
3. Calculate each swimmer’s average speed for the whole race.
4. Whose first m was faster, and how does the graph show it?

Activity 2 — Interpretation Problems (14 min)

Pairs. Every answer must cite a graph feature (“because the line…”).

Graph 1 — Two water tanks. Tank P starts at L and drains steadily to empty at min. Tank Q starts empty and fills steadily to L at min.

1. When do the tanks hold equal amounts, and how much is it?
2. Find each tank’s rate, with sign and units.
3. Write the story of the crossing in one sentence.

Graph 2 — Business costs and income. A start-up’s weekly costs are flat at 600$0$75$ per week.

4. When does the business break even?
5. Shade or name the region of the graph where it makes a loss.
6. What is the profit in week ?

Graph 3 — Two cars, one journey. Car X leaves at 9:00 travelling steadily at km/h. Car Y leaves the same place at 9:30 travelling at km/h.

7. Sketch or read both lines: when and where does Y catch X?
8. What feature of the two lines guarantees Y must catch X eventually?

Socratic scaffolding for Q7–8:

Prompt	Purpose
At 9:30, where is each car?	X: km along. Y: at the start.
How fast does the gap close?	km/h.
So how long to close km?	hour — Y catches X at 10:30.
How far along?	km ✓ (X: ✓).
Which feature guarantees the catch?	Y’s line is steeper; a steeper line from behind must eventually meet a shallower one.
Looking back	The crossing answers “when and where” simultaneously — one point, two coordinates, both meaningful.

Answers: 1. At min, L each; 2. P: L/min; Q: L/min; 3. E.g. “At minutes the draining tank passes the filling tank, each holding L.”; 4. — week ; 5. Left of week , where the income line is below the cost line; 6. 30090$ km from the start; 8. Steeper (faster) line starting behind.

Activity 3 — Inquiry: what the Graph Cannot Tell You (8 min)

Pairs.

The swimmers’ graph (Activity 1) shows B fast early, fading; A steady, winning.

1. List three questions this graph answers.
2. List three questions about the race it cannot answer.
3. One student concludes: “B needs to train endurance.” Fair conclusion or over-reach?

Discussion targets: Answers — who won, by how much, speeds per section, overtaking time. Cannot answer — stroke used, effort, water conditions, whether this race is typical, why B faded. The training conclusion is a plausible hypothesis the graph suggests but cannot establish from one race — distinguishing shows from suggests is the lesson’s capstone. (Connects to L74–77’s assumption discipline.)

Checks for Understanding

(5 minutes — exit ticket, with a printed costs/income graph: costs flat 240$40$ per week)

1. Read the break-even week from the graph, then verify by calculation.
2. What is the loss in week ?
3. What is the profit in week ?
4. What does the vertical gap between the lines represent?
5. Reasoning. The graph suggests income will keep rising forever. Why is this not guaranteed?

Answers: 1. Week (); 2. 120400 - 240 = $160$ profit; 4. Profit (income above costs) or loss (below) that week; 5. The line is an extrapolated trend, not a law — demand, prices or competition can change; the graph records, it does not promise.

Common Misconceptions

Misconception	How to pre-empt it
Reading a crossing as “same speed” rather than “same value”.	Tanks and cars both: at crossing, amounts/positions are equal; the rates plainly differ.
Claiming things the graph does not plot.	The warmup’s “can’t tell” category; the inquiry’s shows/suggests split.
Computing a rate from a curved or flat section as if straight.	Rates from straight sections only at this level; name the restriction.
Sign-blind rates (” L/min” for the draining tank).	Require sign and units: L/min.
Believing the steeper line is “winning”.	Steeper is faster now; higher is ahead. Car Y is steeper but behind until 10:30.
Trusting extrapolation.	Exit Q5 and L83’s population question — assumption must be stated.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Two candles of equal height burn: one lasts hours, the other . Sketch both height–time lines. When is the slow one exactly twice the height of the fast one?

Answer

Heights: and . Twice when , so , giving , hours. (Check: heights and ✓)

E2 (AMC Junior style). A graph shows income 50$180 + $20$ per week. Find the break-even week by calculation, and say what the graph would show there.

Answer

. The lines cross at week , both at 300$.

E3 (Challenge). Cyclist P rides at km/h. Cyclist Q starts km behind P at the same moment, riding at km/h. From their distance–time graphs (same axes, Q measured from Q’s start), when does Q draw level, and how far has each ridden?

Answer

Gap closes at km/h: hours. Q rides km; P rides km (and started km ahead: ✓).

E4 (Challenge). A graph shows two lines crossing twice. Invent a realistic pair of quantities that could do this, and tell the story of both crossings.

Answer

E.g. two runners where A sprints early (leads), B overtakes mid-race (first crossing), A finishes fast and re-passes (second crossing). Or heating costs vs solar output across a year. Any story needing lead changes twice.

Homework

1. A gym charges 0$18$60$12618$ months.
2. From a distance–time graph: a hiker covers km in the first h, rests min, then km in the final h. (a) Average speed in each phase. (b) Average speed overall. (c) Why is (b) not the average of the two moving speeds?
3. Tanks: R starts at L draining at L/min; S starts at L filling at L/min. (a) When do they hold equal amounts? (b) How much is in each then?
4. A lemonade stand’s income is 3$2130$ cups.
5. Reasoning. Two lines on a graph never cross. What does this mean for the quantities, and give a realistic example.
6. Reasoning. From one week’s sales graph, a manager concludes “Fridays are always our best day.” Assess the reasoning.
7. Challenge. Car M travels at km/h; car N leaves minutes later at km/h. When and where does N catch M? Show the graphical reasoning and the calculation.

Answers: Q1 — (b) months, both 1806108 < 13218276 < 324402.\overline{6}124= 3600 - 10t = 150 + 5t \Rightarrow t = 30300790 - 21 = $6960203200300380 \times 3.75 = 300300345$ min after M left.