083i - Exploring Graphs of Real Data

Lesson 83 — Exploring Graphs of Functions from Authentic Data

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

Learning Intentions

• To read information accurately from graphs of real data.
• To understand what the axes, scale and shape of a graph communicate.

Success Criteria

I can:

1. Identify what each axis represents, including units.
2. Read values from a graph, interpolating between marked points.
3. Describe what the overall shape of a graph shows.
4. Spot misleading features of a graph.

Warmup

(6 minutes — one graph, many questions; projected)

Project a simple line graph: temperature in the schoolyard from 6 am to 6 pm, rising from C at 6 am to a peak of C at 2 pm, falling to C by 6 pm.

1. What is on each axis? What are the units?
2. What was the temperature at 10 am? (Read: about C.)
3. When was it C? (Careful — two answers: about noon, and about 4:30 pm.)
4. When was it warming fastest?

Key teaching point from Q3: a horizontal line at C crosses the curve twice. Reading a graph backwards can give several answers, and all must be reported.

Activities

Activity 1 — Explicit Instruction: how to Read a Graph (12 min)

The reading protocol — model on the temperature graph:

1. Title and axes first. What is being measured, against what, in what units?
2. Scale check. What does one gridline step represent on each axis? (They are rarely the same.)
3. Read forwards (given , find ): ruler vertically up from the -value, then across.
4. Read backwards (given , find ): across then down — checking for multiple crossings.
5. Describe the shape last — rising, falling, steep, flat, peaks and troughs.

Vocabulary to establish (used all week): increasing, decreasing, constant, steep (fast change), gradual (slow change), peak, trough.

I do — narrate a full reading of a second graph: water depth in a harbour over 24 hours (tidal, two peaks). Model each protocol step aloud, including the double-crossing when asked “when was the depth 3 m?” (Four times — tides rise and fall twice daily.)

We do: From the tide graph:

1. What was the depth at 9 am?
2. What was the maximum depth, and when?
3. For how long was the depth above 4 m?
4. Is “the tide rose all morning” a fair description? Justify from the graph.

Activity 2 — Graph Reading Circuit (14 min)

Pairs. Three station graphs, printed large. Each has a question set requiring forward reads, backward reads, and one interpretation.

Graph A — Mobile phone battery ( vs hours since 7 am): starts , falls gently, drops steeply during gaming 3–4 pm, then charges rapidly after 9 pm.

1. Battery at noon?
2. When did it hit ?
3. Identify when the phone was being charged, and how you can tell.
4. During which hour did the battery drain fastest? What might explain it?

Graph B — Height of a netball shot (metres vs seconds): rises to a peak of about m at s, falls to hoop height m at about s.

5. Maximum height, and when?
6. Height at s?
7. When was the ball at m (hoop height)? (Two times — on the way up and down.)
8. How long was the ball above m?

Graph C — Population of a country town (people vs years 1990–2020): gradual decline 1990–2005, sharp rise 2005–2012 (mine opens), plateau after.

9. Population in 2000?
10. When did the population first pass ?
11. Describe the three phases of the graph in words.
12. Predict the population in 2025, and state the assumption your prediction makes.

Teacher note on Q12: any extrapolation assumes the current trend continues — the plateau suggests stability, but the graph cannot tell us about the future. This seeds Lesson 86’s discussion of the limits of graphs.

Activity 3 — Inquiry: the Misleading Graph (12 min)

Pairs. Two versions of the same data.

The same juice-sales data (, , , bottles across four weeks) is plotted twice:

• Graph 1: -axis from to .

• Graph 2: -axis from to (truncated).

1. Which graph makes sales look like they are soaring?
2. Are either of the graphs false?
3. When might a company prefer Graph 2? When might Graph 1 be the fairer choice?
4. What should a careful reader always check first?

Socratic scaffolding:

Prompt	Purpose
Describe the visual impression of each.	Graph 1: gentle rise. Graph 2: dramatic climb.
Compute the actual change.	bottles on a base of — a rise.
Is either graph lying about the numbers?	No — both plot the same points accurately.
So where does the different impression come from?	The truncated axis stretches the visible change.
Is truncation always wrong?	No — it can reveal small but real variation. It misleads when the baseline matters and is hidden.
The reader’s defence?	Check the axis start value before reacting to the shape.

Answers: 1. Graph 2; 2. No — both are accurate plots; 3. Graph 2 flatters growth (marketing); Graph 1 gives honest proportion (reporting); 4. Where the axes start and what one step represents.

Checks for Understanding

(6 minutes — exit ticket, using a printed graph of a cyclist’s distance vs time)

The graph shows: steady rise 0–20 min (0 to 6 km), flat 20–30 min, steeper rise 30–45 min (6 to 12 km).

1. How far had the cyclist travelled after minutes?
2. What happened between and minutes? How does the graph show it?
3. When was the cyclist riding fastest? How can you tell?
4. When had the cyclist travelled km?
5. Reasoning. What should you check before comparing two graphs side by side?

Answers: 1. km (steady km/min); 2. A rest — distance stays constant, so the line is flat; 3. – min — the line is steepest ( km/min); 4. km falls in the final phase: gives min; 5. That both use the same scales and axis ranges — otherwise steepness comparisons deceive.

Common Misconceptions

Misconception	How to pre-empt it
Reading the graph as a picture of the event (the netball graph as the ball’s path).	Ask what the axes actually are: height vs time, not height vs horizontal distance. Name this trap explicitly — it recurs in L84.
Ignoring units and scale, reading gridlines as “1 each”.	Protocol step 2 — scale check before any reading.
Reporting one answer to a backwards reading with multiple crossings.	Warmup Q3 and hoop-height Q7 both force the double answer.
Believing a flat line means “nothing is happening”.	The rest stop: the quantity is unchanged, which is itself information.
Treating a truncated axis as dishonest by definition.	The inquiry distinguishes misleading use from legitimate use.
Extrapolating beyond the data with confidence.	Circuit Q12 requires the assumption to be stated.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A graph shows a car’s distance rising steadily by km every hour for hours. How far does it travel, and what does the steepness represent?

Answer

km; the steepness (gradient) is the speed, km/h.

E2 (AMC Junior style). A bath fills for minutes to L, sits for minutes, then drains in minutes. Sketch depth vs time and label the three phases with their rates.

Answer

Rising line ( L/min), flat line ( L/min), falling line ( L/min) — the drain is steeper than the fill.

E3 (Challenge). A graph’s -axis runs from to . The plotted line falls from to . A headline reads “Massive collapse!” Assess the claim.

Answer

The fall is on a base of — about . The truncated axis (a -unit window) makes a tiny change fill the frame. The claim is unsupported.

E4 (Challenge). Two runners’ distance–time graphs cross at min. What does the crossing point mean, and who is winning just before it?

Answer

At the crossing they have covered equal distances — the trailing runner draws level. Just before it, whoever’s curve is higher is ahead. (Crossing does not mean equal speeds — a common misread.)

Homework

1. Using the cyclist graph from class (redrawn on the sheet): (a) distance at min (b) when km was reached (c) average speed in the final phase.
2. A graph shows a kettle’s water temperature: C at s rising to C at s, then constant. (a) Temperature at s (assume a steady rise)? (b) When did it reach C? (c) Why does the graph flatten?
3. Sketch a graph of the depth of water in a plain cylindrical glass being filled at a constant rate. Label axes.
4. Sketch the battery graph of your own (or a family) phone across a typical day, marking charging and heavy-use periods.
5. The same data is plotted with -axis – and -axis –. Describe how each looks, and give one honest use for each version.
6. Reasoning. A distance–time graph is flat from to . A student says the object “went backwards”. Correct them.
7. Reasoning. Explain why “when was the height m?” can have two answers for a thrown ball but “what was the height at s?” has only one.
8. Challenge. A graph shows total rainfall for the year accumulating month by month. What does a steep section mean? Can this graph ever go down? Why?

Answers: Q1 — (a) km (b) min (c) km in min km/h. Q2 — (a) C (b) s (c) water boils at C and cannot exceed it. Q6 — flat means stationary: distance is unchanged; going backwards would need total distance to decrease, impossible on a distance-travelled graph. Q7 — height repeats (up and down) but each instant has exactly one height: time-to-height is many-to-one, height-to-time can be one-to-many. Q8 — steep = wet month; never down — accumulation cannot shrink.