AC9M7A04 • Year 7 Maths • Algebra • Learn
describe relationships between variables represented in graphs of functions from authentic data.
We are learning to describe relationships between variables represented in graphs of functions from authentic data.
A graph of an authentic function shows how a dependent variable changes with an independent variable; axes, scale and units determine the meaning of every point.
Interpret segments through rate of change: rising, falling and horizontal sections represent positive, negative and zero rates, and steepness compares the size of those rates.
Graph evidence can support a relationship or prediction but does not by itself establish causation. Verify claims by reading coordinates and recomputing rise divided by run.
Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.
Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.
Example 1
Final answer: The traveller was stationary for 1 hour, from hour 2 to hour 3.
Check: Both endpoints of the flat segment have distance 8 km.
Example 2
Final answer: The final rising segment is faster: 6 km/h compared with 4 km/h.
Check: The steeper segment has the greater rise per hour on the same axes and scales.
Example 3
Problem: A distance–time graph rises from (0,0) to (2,6), stays flat to (4,6), then rises to (6,14). Describe what each segment means and identify which moving segment is steeper.
Final answer: From 0–2 h the object moves 6 km; from 2–4 h it is stationary; from 4–6 h it moves 8 km. The last moving segment is steeper because its rate is 4 km/h versus 3 km/h.
Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Example 4
Problem: A distance–time graph plots (0,0), (1,4), (2,8), (3,8) and (4,14), with time in hours and distance in kilometres. Describe every interval rate and explain whether the graph alone justifies a distance prediction at hour 5.
Final answer: Across the displayed one-hour intervals the rates are 4, 4, 0 and 6 km/h; because the rate changes and the observed domain ends at hour 4, the graph alone does not justify one particular distance at hour 5.
Check: The complete model for “A distance–time graph plots (0,0), (1,4), (2,8), (3,8) and (4,14), with time in hours and distance in kilometres. Describe every interval rate and explain whether the graph alone justifies a distance prediction at hour 5.” shows Across the displayed one-hour intervals the rates are 4, 4, 0 and 6 km/h; because the rate changes and the observed domain ends at hour 4, the graph alone does not justify one particular distance at hour 5.
Common mistake: Axes read without units.
Correction: State the quantities and scales.
Common mistake: Steepness compared across different axis scales.
Correction: Check scale before judging rate.
Common mistake: Graph described point by point only.
Correction: Summarise intervals and trends.
Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.
1. Which axis usually displays time in a time-dependent graph?
Answer: The horizontal axis.
Hint: Locate the independent variable.
Why: Using the method “Locate the independent variable.” gives The horizontal axis.
2. What does a horizontal segment on a distance–time graph mean?
Answer: Distance did not change, so the object was stationary.
Hint: Compare the segment endpoints.
Why: Using the method “Compare the segment endpoints.” gives Distance did not change, so the object was stationary.
3. A graph rises 6 metres in 3 seconds. Find its rate.
Answer: 2 m/s.
Hint: Divide vertical change by horizontal change.
Why: Using the method “Divide vertical change by horizontal change.” gives 2 m/s.
4. Temperature falls from 18°C to 12°C in 2 hours. State the average hourly change.
Answer: −3°C per hour.
Hint: Calculate (12−18)÷2.
Why: Using the method “Calculate (12−18)÷2.” gives −3°C per hour.
5. On two distance–time graphs with the same axis quantities, units and scales, what does a steeper rising segment show?
Answer: A greater speed.
Hint: Compare rise per unit run only after confirming the axes and scales match.
Why: Using the method “Compare rise per unit run only after confirming the axes and scales match.” gives A greater speed.
6. A line passes through (0,5) and (4,13). Describe its start and rate.
Answer: It starts at 5 and increases by 2 per x-unit.
Hint: Find the intercept and (13−5)÷4.
Why: Using the method “Find the intercept and (13−5)÷4.” gives It starts at 5 and increases by 2 per x-unit.
7. Can a graph of temperature and ice-cream sales prove temperature causes sales?
Answer: No. The graph shows an association, but it does not control other influences or establish which variable caused the change, so a causal conclusion is unsupported.
Hint: Consider other variables and study design.
Why: The reasoning for “Can a graph of temperature and ice-cream sales prove temperature causes sales?” is complete because No. The graph shows an association, but it does not control other influences or establish which variable caused the change, so a causal conclusion is unsupported.
8. A tank rises 20 L in 5 min, stays flat 2 min, then falls 9 L in 3 min. Give all rates.
Answer: The rates are 20÷5=+4 L/min, 0÷2=0 L/min and −9÷3=−3 L/min; the signs show filling, no change and draining.
Hint: Calculate change divided by time for each segment.
Why: The complete model for “A tank rises 20 L in 5 min, stays flat 2 min, then falls 9 L in 3 min. Give all rates.” shows The rates are 20÷5=+4 L/min, 0÷2=0 L/min and −9÷3=−3 L/min; the signs show filling, no change and draining.
9. A distance–time graph plots (0,0), (1,4), (2,8), (3,8) and (4,14), with time in hours and distance in kilometres. Describe every interval rate and explain whether the graph alone justifies a distance prediction at hour 5.
Answer: Across the displayed one-hour intervals the rates are 4, 4, 0 and 6 km/h; because the rate changes and the observed domain ends at hour 4, the graph alone does not justify one particular distance at hour 5.
Hint: Calculate each consecutive rise per one-hour run, then consider the observed domain and changing rate.
Why: The complete model for “A distance–time graph plots (0,0), (1,4), (2,8), (3,8) and (4,14), with time in hours and distance in kilometres. Describe every interval rate and explain whether the graph alone justifies a distance prediction at hour 5.” shows Across the displayed one-hour intervals the rates are 4, 4, 0 and 6 km/h; because the rate changes and the observed domain ends at hour 4, the graph alone does not justify one particular distance at hour 5.
10. A distance–time graph rises from (0,0) to (2,6), stays flat to (4,6), then rises to (6,14). Describe what each segment means and identify which moving segment is steeper.
Answer: From 0–2 h the object moves 6 km; from 2–4 h it is stationary; from 4–6 h it moves 8 km. The last moving segment is steeper because its rate is 4 km/h versus 3 km/h.
Hint: Represent the information first, then calculate, interpret and independently check the result.
Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Content description: describe relationships between variables represented in graphs of functions from authentic data.
The Australian Curriculum code above is exact. Victorian Year 7, NSW Stage 4, US Grade 7, England Key Stage 3, New Zealand Level 4 and comparable international curricula contain broadly related learning, but code-to-code equivalence varies by jurisdiction.
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Before you watch:
Khan Academy — Interpret the relationship between two measured quantities from a graph.
As you watch: What do the coordinates of a point mean in the situation being modelled?
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Try it: Sketch a graph for a tap delivering 2 litres each minute, label both axes and explain what (3, 6) represents.
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Mapped skill: describe relationships between variables represented in graphs of functions from authentic data
These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M7A04 · Year 7 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M7A04 · Level 7 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA4-LIN-C-01 · Stage 4 |
| United States (USA) | Common Core State Standards for Mathematics | Grade 7 |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 7 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 8, Key Stage 3 |
| India | NCERT / CBSE — Mathematics | Class 7 |
Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.
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Topic reference: AC9M7A04 — Relationships Between Variables in Authentic Graphs
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