MYP 3 · Maths
INTERPRETING TABLES AND GRAPHS
60 questions across 4 sub-topics
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Interpreting tables
Interpreting graphs
Line graphs
Travel graphs
Interpreting tables 12 questions
QUESTION 1
5 marks
Criterion A
Medium
| Month | Jan | Feb | Mar | Apr | May |
|---|---|---|---|---|---|
| Rainfall (mm) | 85 | 62 | 78 | 95 | 110 |
The table shows monthly rainfall (mm) recorded at a weather station.
| Month | Jan | Feb | Mar | Apr | May |
|---|---|---|---|---|---|
| Rainfall (mm) | 85 | 62 | 78 | 95 | 110 |
a.
State the rainfall in March.
[1] b.
Find the total rainfall over the 5 months.
[2] c.
Find the difference between the wettest and driest months shown.
[2] Show complete worked solution
(a)
78 mm
(b)
$$85+62+78+95+110=430 \text{ mm}$$
(c)
Wettest: May (110 mm). Driest: Feb (62 mm). Difference $=110-62=48$ mm.
QUESTION 2
6 marks
Criterion A
Medium
| Bus | Car | Walk | Total | |
|---|---|---|---|---|
| Grade 9 | 18 | 24 | 8 | 50 |
| Grade 10 | 15 | 30 | 5 | 50 |
| Total | 33 | 54 | 13 | 100 |
The two-way table shows how students in two grades travel to school.
| Bus | Car | Walk | Total | |
|---|---|---|---|---|
| Grade 9 | 18 | 24 | 8 | 50 |
| Grade 10 | 15 | 30 | 5 | 50 |
| Total | 33 | 54 | 13 | 100 |
a.
How many Grade 10 students travel by car?
[1] b.
What percentage of ALL 100 students walk to school?
[2] c.
Which grade has a higher proportion of bus travellers, and by how many percentage points?
[3] Show complete worked solution
(a)
30 students
(b)
$$\frac{13}{100}\times100=13\%$$
(c)
Grade 9: $\frac{18}{50}\times100=36\%$. Grade 10: $\frac{15}{50}\times100=30\%$. Grade 9 is higher, by $36-30=6$ percentage points.
QUESTION 3
6 marks
Criterion B
Medium
| Year | 2019 | 2020 | 2021 | 2022 | 2023 |
|---|---|---|---|---|---|
| Sales (thousands of dollars) | 210 | 195 | 230 | 265 | 310 |
Investigate the year-on-year percentage change in a company's sales.
| Year | 2019 | 2020 | 2021 | 2022 | 2023 |
|---|---|---|---|---|---|
| Sales (thousands of dollars) | 210 | 195 | 230 | 265 | 310 |
a.
Calculate the percentage change in sales from 2019 to 2020, and from 2022 to 2023.
[3] b.
Calculate the percentage change for every other consecutive pair of years, and describe the overall pattern in the company's growth rate.
[3] Show complete worked solution
(a)
2019 to 2020: $\frac{195-210}{210}\times100=-7.14\%$ (a decrease). 2022 to 2023: $\frac{310-265}{265}\times100=16.98\%$ (an increase).
(b)
2020 to 2021: $\frac{230-195}{195}\times100=17.9\%$. 2021 to 2022: $\frac{265-230}{230}\times100=15.2\%$. After the initial dip, the company shows strong, fairly consistent double-digit percentage growth each year from 2020 onward.
QUESTION 4
4 marks
Criterion C
Medium
| Bus | Car | Walk | Total | |
|---|---|---|---|---|
| Grade 9 | 18 | 24 | 8 | 50 |
| Grade 10 | 15 | 30 | 5 | 50 |
| Total | 33 | 54 | 13 | 100 |
A classmate reads the two-way table above (student travel to school) and claims: 'more Grade 10 students walk than Grade 9 students, since walking is listed as only 5 for Grade 10 vs 8 for Grade 9... wait, that's backwards from what I said.'
a.
Correct the classmate's confused statement: which grade actually has MORE students walking, using the table?
[1] b.
Explain a reliable strategy for reading a two-way table accurately: how do you find the right row and column intersection for a specific data point (e.g. 'Grade 10 students who walk')?
[3] Show complete worked solution
(a)
Grade 9 has more students walking (8) compared to Grade 10 (5).
(b)
Find the row labelled with the category you want (e.g. 'Grade 10'), then move along that row until you reach the column labelled with the second category (e.g. 'Walk') — the value at that exact row-column intersection is the answer. Reading carelessly or mixing up rows and columns is a common source of errors.
QUESTION 5
5 marks
Criterion D
Medium
| Item | Rent | Food | Transport | Savings | Other |
|---|---|---|---|---|---|
| Monthly cost (dollars) | 950 | 420 | 180 | 250 | 200 |
A family's monthly budget is shown in the table.
| Item | Rent | Food | Transport | Savings | Other |
|---|---|---|---|---|---|
| Monthly cost (dollars) | 950 | 420 | 180 | 250 | 200 |
a.
Find the family's total monthly spending (all categories combined).
[2] b.
The family's monthly income is \$2,200. Are they spending within their means? If they want to increase Savings to \$400 without changing income, which OTHER category(s) would need to be reduced, and by how much in total?
[3] Show complete worked solution
(a)
$$950+420+180+250+200=2000$$
(b)
Yes, they currently spend \$2000 of \$2200 income (\$200 left over). To increase Savings from \$250 to \$400 (a \$150 increase) while keeping total spending at or under \$2200, they could use their current \$200 surplus — no other category needs cutting, since $2000+150=2150 \le 2200$.
QUESTION 6
6 marks
Criterion A
Medium
| City | Tokyo | London | Cairo | Nairobi | Oslo | Reykjavik |
|---|---|---|---|---|---|---|
| Avg. daily solar hours | 5.8 | 3.4 | 9.4 | 6.9 | 3.9 | 3.0 |
The table shows average daily solar sunshine hours in 6 cities, used for a renewable energy feasibility study.
| City | Tokyo | London | Cairo | Nairobi | Oslo | Reykjavik |
|---|---|---|---|---|---|---|
| Avg. daily solar hours | 5.8 | 3.4 | 9.4 | 6.9 | 3.9 | 3.0 |
a.
Which city has the most solar potential, and by how many hours per day does it exceed the LEAST sunny city?
[2] b.
Find the mean daily sunshine hours across all 6 cities, to 2 decimal places.
[2] c.
A solar company requires AT LEAST 5.5 average daily hours to consider a location commercially viable. List which cities qualify.
[2] Show complete worked solution
(a)
Cairo (9.4 hours) has the most. Excess over Reykjavik (3.0, the least): $9.4-3.0=6.4$ hours.
(b)
$$\frac{5.8+3.4+9.4+6.9+3.9+3.0}{6}=\frac{32.4}{6}=5.40 \text{ hours}$$
(c)
Tokyo (5.8) and Cairo (9.4) and Nairobi (6.9) qualify — the three cities with values above 5.5.
QUESTION 7
5 marks
Criterion A
Medium
| Under 18 | 18-40 | Over 40 | Total | |
|---|---|---|---|---|
| Vaccinated | 420 | 680 | 510 | 1610 |
| Not vaccinated | 180 | 140 | 70 | 390 |
| Total | 600 | 820 | 580 | 2000 |
A public health department's vaccination data for 2,000 residents is shown in the two-way table.
| Under 18 | 18-40 | Over 40 | Total | |
|---|---|---|---|---|
| Vaccinated | 420 | 680 | 510 | 1610 |
| Not vaccinated | 180 | 140 | 70 | 390 |
| Total | 600 | 820 | 580 | 2000 |
a.
What percentage of the 18-40 age group is vaccinated?
[2] b.
Which age group has the LOWEST vaccination rate (as a percentage of its own group), and what is that rate?
[3] Show complete worked solution
(a)
$$\frac{680}{820}\times100\approx82.9\%$$
(b)
Under 18: $\frac{420}{600}\times100=70\%$. 18-40: $\approx82.9\%$ (from part a). Over 40: $\frac{510}{580}\times100\approx87.9\%$. The Under 18 group has the LOWEST rate, at 70%.
QUESTION 8
4 marks
Criterion A
Hard
| Year | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
|---|---|---|---|---|---|---|
| EV sales (1000s) | 45 | 68 | 102 | 178 | 295 | 412 |
The table shows annual electric vehicle (EV) sales (in thousands) in a country over 6 years.
| Year | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
|---|---|---|---|---|---|---|
| EV sales (1000s) | 45 | 68 | 102 | 178 | 295 | 412 |
a.
Find the year-on-year PERCENTAGE growth for each consecutive pair of years.
[4] Show complete worked solution
(a)
2018 to 2019: $\frac{68-45}{45}\times100\approx51.1\%$. 2019 to 2020: $\frac{102-68}{68}\times100\approx50.0\%$. 2020 to 2021: $\frac{178-102}{102}\times100\approx74.5\%$. 2021 to 2022: $\frac{295-178}{178}\times100\approx65.7\%$. 2022 to 2023: $\frac{412-295}{295}\times100\approx39.7\%$.
QUESTION 9
6 marks
Criterion B
Hard
Investigate whether the growth PATTERN in EV sales (from the table above) is accelerating, decelerating, or roughly constant, using the percentage growth rates you calculated.
| Year | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
|---|---|---|---|---|---|---|
| EV sales (1000s) | 45 | 68 | 102 | 178 | 295 | 412 |
a.
List the 5 year-on-year percentage growth rates in order (51.1%, 50.0%, 74.5%, 65.7%, 39.7%), and describe whether they show a clear, consistent trend, or a more complex pattern.
[3] b.
Based on this pattern, would it be reasonable to assume EV sales will KEEP growing at over 50% per year indefinitely? Justify your answer using the recent trend in the data.
[3] Show complete worked solution
(a)
The rates don't show a simple consistent trend — they rise (51.1%, 50.0%, 74.5%) then fall (74.5%, 65.7%, 39.7%), suggesting the growth RATE itself peaked around 2020-2021 and has since been slowing, even though sales in absolute terms are still increasing every year.
(b)
No — the most RECENT data (2022 to 2023 growth of only 39.7%, down from a peak of 74.5%) suggests the growth rate is DECELERATING, not sustaining indefinitely. This is a common real-world pattern for emerging technology adoption (rapid initial growth eventually slows as the market matures), so extrapolating the early high growth rates far into the future would likely overestimate future sales.
QUESTION 10
5 marks
Criterion C
Medium
Investigate what information is LOST when a detailed data table is summarized down to just its mean value, using the vaccination data table from earlier.
a.
Calculate the OVERALL vaccination rate across all 2,000 residents (not broken down by age group).
[2] b.
Compare this single overall figure (80.5%) to the THREE separate age-group rates found earlier (70%, 82.9%, 87.9%). Explain what important information a policy-maker would MISS if they were only given the single 80.5% figure, without the age-group breakdown.
[3] Show complete worked solution
(a)
$$\frac{1610}{2000}\times100=80.5\%$$
(b)
A policy-maker seeing only 80.5% might assume vaccination coverage is fairly uniform, missing that the UNDER-18 group is significantly BEHIND (70%) compared to older groups (up to 87.9%) — this detail matters because it identifies a SPECIFIC group that may need targeted outreach or a different vaccination strategy, information that's completely hidden by a single averaged summary statistic.
QUESTION 11
3 marks
Criterion C
Medium
| Product | Jan Sales | Feb Sales | Mar Sales |
|---|---|---|---|
| Widget A | 340 | 28 | 355 |
| Widget B | 210 | 225 | 198 |
A store manager reviews a sales table and notices something odd about Widget A's February figure.
| Product | Jan Sales | Feb Sales | Mar Sales |
|---|---|---|---|
| Widget A | 340 | 28 | 355 |
| Widget B | 210 | 225 | 198 |
a.
Explain what appears SUSPICIOUS about Widget A's February sales figure (28) compared to its January (340) and March (355) figures, and suggest a likely explanation (e.g. a data entry error) rather than assuming it reflects a genuine, real sales collapse.
[3] Show complete worked solution
(a)
28 is drastically lower than both neighbouring months (340 and 355) — a sudden 92%+ drop followed by an immediate, complete recovery to normal levels the very next month is highly unusual for typical sales patterns. A far more likely explanation is a DATA ENTRY ERROR (e.g. a missing digit — perhaps '280' was mistyped as '28', or a decimal/typo issue), rather than a genuine one-month sales collapse with no lasting cause.
QUESTION 12
7 marks
Criterion D
Hard
| Region | Population (millions) | Area (km²) |
|---|---|---|
| Region A | 4.2 | 850 |
| Region B | 2.8 | 3200 |
| Region C | 6.5 | 1450 |
An urban planner compares 3 regions using population and land area data.
| Region | Population (millions) | Area (km²) |
|---|---|---|
| Region A | 4.2 | 850 |
| Region B | 2.8 | 3200 |
| Region C | 6.5 | 1450 |
a.
Calculate the population DENSITY (people per km²) for each region, and identify which region is most densely populated.
[3] b.
A politician claims Region C 'has the biggest population problem' since it has the most PEOPLE. Evaluate this claim using density rather than raw population, and explain why density might be a more useful metric for planning infrastructure like housing or transport.
[4] Show complete worked solution
(a)
Region A: $\frac{4.2\text{ million}}{850}\approx4941$ people/km². Region B: $\frac{2.8\text{ million}}{3200}=875$ people/km². Region C: $\frac{6.5\text{ million}}{1450}\approx4483$ people/km². Region A is most densely populated.
(b)
While Region C does have the highest RAW population (6.5 million), Region A actually has the highest DENSITY (approximately 4941 people/km², compared to Region C's approximately 4483). Density is often more relevant for infrastructure planning because it reflects how CROWDED an area actually is — a region with a large population spread over a huge area may have plenty of room to expand housing/transport, while a smaller-population but highly dense region may face more acute overcrowding pressures per square kilometre, even with fewer total residents.
Interpreting graphs 13 questions
QUESTION 1
5 marks
Criterion A
Medium
The graph shows the number of visitors to a museum each day over one week.
a.
On which day were there the most visitors? State the number.
[1] b.
Find the total number of visitors across the whole week.
[2] c.
Find the mean (average) number of visitors per day, correct to 1 decimal place.
[2] Show complete worked solution
(a)
Saturday, with 90 visitors.
(b)
$$45+62+38+71+55+90+84=445$$
(c)
$$445 \div 7 \approx 63.6 \text{ visitors}$$
QUESTION 2
3 marks
Criterion A
Medium
The graph shows a gym's membership numbers over 6 years.
a.
Between which two consecutive years did membership DECREASE?
[1] b.
Find the overall percentage change in membership from 2018 to 2023.
[2] Show complete worked solution
(a)
Between 2019 and 2020 (from 135 down to 98).
(b)
$$\frac{175-120}{120}\times100 \approx 45.8\% \text{ increase}$$
QUESTION 3
4 marks
Criterion B
Medium
Investigate how the STEEPNESS of a line graph relates to how quickly a quantity is changing.
a.
Using the graph, find the increase in bacteria count during Hour 1-2 (steep section) and during Hour 0-1 (flatter section).
[2] b.
Compare the two increases you found. What does a STEEPER section of the graph tell you about the rate of change?
[2] Show complete worked solution
(a)
Hour 0-1: $15-10=5$ (thousand). Hour 1-2: $40-15=25$ (thousand).
(b)
The steeper section (Hour 1-2) shows a much bigger increase (25,000) than the flatter section (5,000) over the same 1-hour period — a steeper line always means the quantity is changing FASTER (a higher rate of change).
QUESTION 4
4 marks
Criterion C
Medium
A classmate looks at a graph with a very steep RISING line at the start and a nearly flat line at the end, and says 'the graph shows things are getting worse near the end, since the line is barely moving.'
a.
Explain why the classmate's interpretation might be incorrect, IF the quantity being measured (e.g. distance travelled, or population) is one where flat simply means 'staying the same', not 'getting worse'.
[2] b.
State what a downward-sloping section of a graph would represent instead, contrasting it with the flat section.
[2] Show complete worked solution
(a)
A flat (or nearly flat) section of a graph means the quantity has STOPPED CHANGING (it's constant), not that it's declining or 'getting worse' — 'worse' would require the line to actually go DOWNWARDS, which is different from just being flat.
(b)
A downward-sloping section means the quantity is actually DECREASING over time — this is genuinely different from a flat section (no change) and should not be confused with it just because both might look 'less exciting' than a steep rise.
QUESTION 5
5 marks
Criterion D
Medium
A small business owner tracks monthly revenue, shown in the graph.
a.
Identify the month with the lowest revenue, and suggest one possible real-world reason for a dip at that point (using general business knowledge, not specific data given).
[2] b.
The owner wants to know if the business is trending upward overall. Compare the first 3 months' average to the last 3 months' average to support your answer.
[3] Show complete worked solution
(a)
March had the lowest revenue (\$2200). A possible reason could be a seasonal slow period, reduced customer spending after the holidays, or a one-off disruption to business.
(b)
First 3 months average: $(2400+2650+2200)/3\approx2416.67$. Last 3 months average: $(2800+3100+2950)/3\approx2950$. Since the second average is notably higher, the business does appear to be trending upward overall, despite the March dip.
QUESTION 6
4 marks
Criterion A
Medium
The graph shows a city's daily water usage (in kilolitres) over one week.
a.
On which day was water usage lowest, and what was the value?
[1] b.
Find the mean daily water usage across the week, and the range.
[3] Show complete worked solution
(a)
Wednesday, 790 kL.
(b)
Mean: $\frac{820+865+790+910+875+940+905}{7}=\frac{6105}{7}\approx872.1$kL. Range: $940-790=150$kL.
QUESTION 7
2 marks
Criterion A
Easy
The graph shows annual deforestation (km²) in a protected region over 5 years, following the introduction of new conservation policies in 2020.
a.
Between which two years did deforestation DECREASE, and by how much?
[2] Show complete worked solution
(a)
Between 2020 and 2021, decreasing from 410 to 285 — a decrease of 125 km².
QUESTION 8
6 marks
Criterion B
Hard
Investigate whether the conservation policy (introduced in 2020) appears to have been EFFECTIVE, using the deforestation graph above.
a.
Compare deforestation in 2019 (before the policy) to 2021 (one year after the policy). Does this single comparison support the policy being effective?
[2] b.
Now examine the FULL trend, including 2022 (520km²) and 2023 (610km²). Does the complete picture still support the same conclusion? Explain what a policy evaluator should be cautious about when judging effectiveness from only 1-2 years of data.
[4] Show complete worked solution
(a)
2019: 340km². 2021: 285km² — LOWER than 2019, which superficially supports the policy being effective.
(b)
No — the complete picture shows deforestation has actually risen SHARPLY since 2021, reaching its HIGHEST level by 2023 (610km², well above even the pre-policy 2019 level of 340km²). This shows why judging a policy's effectiveness from just 1-2 years of data can be misleading: an initial dip might reflect a temporary effect, random fluctuation, or an unrelated factor, rather than genuine long-term policy success — a full, multi-year trend is needed before drawing reliable conclusions.
QUESTION 9
6 marks
Criterion B
Hard
Investigate the relationship between the SHAPE of a distance-time graph and the concept of 'slowing down', using the graph shown (a test trolley's progress).
a.
Calculate the average speed during the first 5 seconds and during the last 5 seconds, from 20 s to 25 s.
[3] b.
Compare these two speeds. Explain what this shows about the trolley's motion and how the change is represented by the shape of the graph.
[3] Show complete worked solution
(a)
From the graph, during the first 5 seconds the distance rises from $0$ m to $15$ m: $$v=\frac{15-0}{5}=\boxed{3.0\text{ m s}^{-1}}.$$ During the last 5 seconds it rises from $44$ m to $45$ m: $$v=\frac{45-44}{25-20}=\boxed{0.20\text{ m s}^{-1}}.$$
(b)
The speed decreases from $3.0\text{ m s}^{-1}$ to $0.20\text{ m s}^{-1}$. The trolley is therefore slowing down. On a distance-time graph, speed is the gradient; the curve becomes less steep toward the end, showing the smaller speed.
QUESTION 10
3 marks
Criterion C
Medium
A classmate looks at a graph with a SHARP spike (a single very tall, narrow peak) in an otherwise smooth dataset, and immediately concludes 'this must be the most important/reliable data point, since it's the most extreme.'
a.
Explain why an isolated extreme spike, especially one that doesn't fit the surrounding pattern, should actually be treated with MORE suspicion (possibly an error or unusual one-off event), not automatically treated as the most 'important' or reliable point.
[3] Show complete worked solution
(a)
An isolated spike that breaks sharply from an otherwise smooth trend often signals a MEASUREMENT ERROR, a one-off unusual event, or a data recording mistake — not necessarily a genuinely more 'important' or reliable data point. Reliable data typically fits a coherent pattern with the surrounding context; a lone extreme value that doesn't connect logically to its neighbours deserves careful scrutiny rather than automatic trust, precisely because it stands out as inconsistent with everything around it.
QUESTION 11
4 marks
Criterion C
Medium
A student is asked to 'describe the trend' in a graph showing temperature rising, then falling, then rising again over a year, and simply writes: 'it goes up and down.'
a.
Explain why this description, while technically not FALSE, fails to communicate meaningful mathematical information about the graph.
[2] b.
Write a genuinely informative one-sentence description of a hypothetical temperature graph that rises steadily from January to July, then falls steadily from July to December, reaching similar starting and ending values.
[2] Show complete worked solution
(a)
'It goes up and down' is vague and could describe almost ANY non-constant graph — it fails to specify WHEN each change happens, the MAGNITUDE of each rise/fall, or WHICH sections are steeper/flatter than others. Good mathematical communication about a graph's trend should identify specific intervals, quantify approximate values or changes, and describe the overall SHAPE precisely enough that someone could roughly reconstruct the graph from the description alone.
(b)
Example: 'Temperature rises steadily from a low in January to a peak around July, then falls at a similar steady rate back down to approximately the same level by December, forming a roughly symmetric seasonal pattern.'
QUESTION 12
6 marks
Criterion D
Medium
A traffic engineer studies vehicle flow through an intersection over a day, planning a new traffic light timing system.
a.
Identify the PEAK traffic period, and the value at that time.
[2] b.
The engineer wants to design light timing that gives MORE green-light time to the busier direction during peak periods, but LESS resource-intensive standard timing during quiet periods. Using the data, identify TWO time periods where a SIMPLER, less resource-intensive timing system could reasonably be used instead of a complex adaptive one, and justify your choice using the traffic volume figures.
[4] Show complete worked solution
(a)
Peak at 2pm, with 124 vehicles/hour.
(b)
8am (12 vehicles/hour) and 6pm (52 vehicles/hour) are both relatively LOW-traffic periods compared to the midday peak (89-124 vehicles/hour) — during these quieter times, a simpler, fixed-timing traffic light system would likely be sufficient, since there's less risk of significant congestion, saving the cost/complexity of running an adaptive system when it's not really needed.
QUESTION 13
7 marks
Criterion D
Hard
An economic development agency tracks jobs created in a manufacturing sector over 5 years, to evaluate a government investment program.
a.
Find the percentage change in jobs created from 2019 to 2020, and suggest a plausible real-world reason for this specific pattern (without being given additional context).
[3] b.
The agency wants to evaluate whether their investment program (assume it started in 2021) appears successful, using the JOBS CREATED trend. Find the percentage growth from 2020 to 2022, and discuss ONE limitation of using this comparison alone to credit the investment program specifically (i.e. what ELSE might explain the recovery, besides the program?).
[4] Show complete worked solution
(a)
$\frac{9800-13200}{13200}\times100\approx-25.8\%$ — a significant decrease. A plausible explanation: 2020 was affected by a major global economic disruption (e.g. a pandemic or recession), which is a well-known real-world factor that caused widespread job losses/hiring freezes across many industries during that period.
(b)
Growth 2020 to 2022: $\frac{18900-9800}{9800}\times100\approx92.9\%$. Limitation: this strong growth could be partly (or largely) due to a NATURAL economic recovery/rebound after the 2020 disruption, rather than being solely caused by the investment program — without a comparison to a SIMILAR region WITHOUT the program (a 'control' comparison), it's difficult to isolate how much of this growth is genuinely attributable to the program itself versus broader economic recovery trends.
Line graphs 13 questions
QUESTION 1
5 marks
Criterion A
Medium
The line graph shows the temperature recorded at a weather station throughout one day.
a.
State the temperature at 12pm.
[1] b.
During which time interval did the temperature rise the fastest?
[2] c.
Find the overall temperature range for the day (highest minus lowest recorded value).
[2] Show complete worked solution
(a)
22°C
(b)
Between 9am and 12pm, the temperature rose from 14°C to 22°C — an increase of 8°C in 3 hours, the steepest rise shown.
(c)
Highest: 26°C (3pm). Lowest: 8°C (6am). Range $=26-8=18°C$.
QUESTION 2
4 marks
Criterion A
Medium
The line graph shows the height of a hot air balloon over time during a flight.
a.
Find the balloon's height at 30 minutes.
[1] b.
Describe what happens to the balloon's height between 30 and 50 minutes, and calculate the rate of descent (in m per minute) over this interval.
[3] Show complete worked solution
(a)
75 m
(b)
The balloon descends from 75 m to 45 m between minute 30 and minute 50 — a decrease of 30 m over 20 minutes. Rate $=30\div20=1.5$ m per minute.
QUESTION 3
4 marks
Criterion B
Medium
Investigate what a horizontal (flat) section of a line graph represents, using a real scenario.
a.
Using the graph (showing water level in a tank being filled), identify the time interval where the graph is flat, and state the water level during that interval.
[2] b.
Suggest a real-world explanation for why the water level might stay constant for 2 hours in the middle of a filling process.
[2] Show complete worked solution
(a)
The graph is flat between Hour 1 and Hour 3, with the water level staying at 45 cm.
(b)
A likely explanation is that the water supply was paused or turned off during that period (e.g. for a scheduled break, a valve being closed, or a technical issue), before filling resumed afterward.
QUESTION 4
4 marks
Criterion C
Medium
A classmate says a line graph and a bar graph 'show exactly the same information, just in a different shape', so it doesn't matter which one you use.
a.
Explain one key advantage of a LINE graph over a bar graph, specifically related to showing change over a CONTINUOUS variable like time.
[2] b.
Explain one situation where a BAR graph might actually be more appropriate than a line graph.
[2] Show complete worked solution
(a)
A line graph directly shows the TREND and rate of change between data points (via the slope of each segment), and can suggest values BETWEEN measured points — a bar graph only shows isolated values at each point, with no visual sense of a continuous trend connecting them.
(b)
A bar graph is more appropriate for comparing separate, unrelated CATEGORIES (like sales by different named products, or votes for different candidates) where there's no meaningful 'in-between' value connecting one category to the next — a line connecting them would be misleading.
QUESTION 5
5 marks
Criterion D
Medium
A person tracking a fitness goal records their weight weekly, shown in the graph.
a.
Find the total weight lost from Week 1 to Week 6.
[2] b.
Find the average weight loss per week over the 6 weeks, and use it to predict the weight at Week 8 if the same average rate continues.
[3] Show complete worked solution
(a)
$$82-65=17 \text{ kg}$$
(b)
Average loss per week $=17\div5=3.4$ kg (over 5 intervals from Week 1 to Week 6). Predicted Week 8 weight: $65 - 2\times3.4 = 58.2$ kg.
QUESTION 6
6 marks
Criterion A
Medium
The line graph shows temperature readings during one day.
a.
State the temperature at 3pm, and identify the time of the maximum temperature.
[2] b.
Find the mean temperature across the 6 readings.
[2] c.
Find the RATE of temperature change (°C per hour) between 12pm and 3pm.
[2] Show complete worked solution
(a)
27°C at 3pm, which is also the maximum.
(b)
$$\frac{14+18+23+27+22+16}{6}=\frac{120}{6}=20°C$$
(c)
$$\frac{27-23}{3}\approx1.33°C\text{/hour}$$
QUESTION 7
3 marks
Criterion A
Medium
The line graph shows the mass (mg) of a radioactive substance remaining over time.
a.
Find the mass remaining after 6 days.
[1] b.
Find the percentage of the ORIGINAL mass (500mg) that remains after 10 days.
[2] Show complete worked solution
(a)
296 mg.
(b)
$$\frac{209}{500}\times100=41.8\%$$
QUESTION 8
5 marks
Criterion B
Hard
Investigate whether the radioactive decay graph above represents CONSTANT decrease (same amount lost per day) or PROPORTIONAL decrease (same PERCENTAGE lost per day).
a.
Find the ACTUAL amount lost (in mg) between day 0-1, and between day 1-2. Are these amounts equal?
[2] b.
Find the PERCENTAGE lost between day 0-1, and between day 1-2. State what pattern this reveals about how radioactive decay actually works.
[3] Show complete worked solution
(a)
Day 0-1: $500-420=80$mg. Day 1-2: $420-353=67$mg. NOT equal — so the loss is not a constant fixed amount each day.
(b)
Day 0-1: $\frac{80}{500}\times100=16\%$. Day 1-2: $\frac{67}{420}\times100\approx16\%$. These percentages ARE approximately equal — revealing that radioactive decay loses a roughly CONSTANT PERCENTAGE of the remaining amount each day (proportional/exponential decay), rather than losing a fixed amount, explaining why the graph curves rather than forming a straight line.
QUESTION 9
6 marks
Criterion B
Hard
Investigate whether TWO different-looking line graphs (one steep-then-flat, one flat-then-steep) could represent the SAME total change over the SAME time period, just distributed differently.
a.
Using the graph shown (steep at first, flattening later), find the TOTAL distance covered over the full 4 hours.
[2] b.
Now imagine a SECOND journey that instead starts flat and gets steep later, described by values $0,2,4,7,52$ at the same hours ($0,1,2,3,4$). Confirm this journey ALSO covers 52km total, then explain what real-world difference in the JOURNEY (not the total distance) this different shape would represent, compared to the first graph.
[4] Show complete worked solution
(a)
52 km total (final value, since start was 0).
(b)
Second journey final value is also 52km, confirming the same TOTAL distance. However, the shapes represent very different journeys: the FIRST graph (steep-then-flat) suggests fast travel early, slowing down later (e.g. starting energetic, then tiring or hitting traffic) — while the SECOND (flat-then-steep) suggests a slow start followed by a fast finish (e.g. a warm-up period followed by a sprint). Same total distance, but very different speed patterns throughout the journey.
QUESTION 10
4 marks
Criterion C
Medium
A classmate presents a line graph of monthly sales with NO axis labels or units, saying 'the numbers speak for themselves.'
a.
Explain why a graph without axis labels or units fails as effective mathematical communication, even if the shape/trend is visually clear.
[2] b.
List the minimum THREE pieces of labelling information every properly communicated graph should include.
[2] Show complete worked solution
(a)
Without axis labels, a reader has NO way to know what quantity is being measured (sales in dollars? units sold? something else?), what time period each point represents, or what SCALE the numbers are on — a visually clear trend is meaningless without this context, since the same shape could represent wildly different real situations depending on the actual units and values involved.
(b)
1) A label for the horizontal (x) axis, including units if relevant. 2) A label for the vertical (y) axis, including units. 3) A title or caption stating what the graph represents overall (and ideally the data source/time period).
QUESTION 11
3 marks
Criterion C
Medium
A student describes a line graph's trend as 'the line goes up a lot', without giving any numbers.
a.
Given the graph shows temperature rising from 14°C at 6am to 27°C at 3pm (over 9 hours), rewrite the student's vague description as a precise, quantified statement including the actual rise and the time period.
[3] Show complete worked solution
(a)
Precise version: 'Temperature rose by $27-14=13°C$ over the 9-hour period from 6am to 3pm, an average increase of approximately $13\div9\approx1.4°C$ per hour.' This version specifies the exact magnitude of change, the time period, and an average rate — far more informative than 'goes up a lot'.
QUESTION 12
6 marks
Criterion D
Hard
An electric vehicle's battery charge level is monitored during charging, shown in the graph.
a.
Find the AVERAGE charging rate (mAh per hour) over the full 8-hour charge.
[2] b.
The vehicle needs at least 550 mAh to complete a planned trip. Estimate (using the graph's trend) approximately how many hours of charging are needed to reach 550 mAh, and discuss why charging RATE often SLOWS as a battery approaches full capacity in real electric vehicles (a well-known real-world charging characteristic), meaning a simple linear estimate might not be perfectly accurate.
[4] Show complete worked solution
(a)
$$600\div8=75 \text{ mAh/hour}$$
(b)
Using the roughly linear trend, estimated hours for 550 mAh: $550\div75\approx7.3$ hours. However, real EV batteries typically charge FASTER when nearly empty and SLOWER as they approach full capacity (to protect battery health) — this means a simple linear extrapolation likely UNDERESTIMATES the true time needed near the end of charging, since the last portion of charge often takes disproportionately longer than the earlier, faster-charging portion.
QUESTION 13
6 marks
Criterion D
Hard
An economist tracks a region's unemployment rate over a year.
a.
Find the month with highest unemployment, and the overall change in rate from January to November.
[2] b.
A government official claims 'unemployment is under control since it decreased every month from July onward.' Evaluate this claim using the DATA, and discuss what additional information (e.g. data BEYOND November, or the underlying causes of the July peak) would help determine whether this is a genuine sustained improvement or a temporary seasonal fluctuation.
[4] Show complete worked solution
(a)
Highest: July (7.1%). Change January to November: $5.0-4.2=0.8$ percentage points increase.
(b)
The claim is partially supported by the data — the rate DID decrease from July (7.1%) through September (6.3%) to November (5.0%). However, evaluating whether this is 'under control' requires more context: is this a recurring SEASONAL pattern (e.g. summer tourism jobs boosting July unemployment temporarily, then a normal autumn recovery), or a genuine structural improvement? Data from PRIOR years (to check for a repeating seasonal pattern) and data BEYOND November (to see if the improvement continues, plateaus, or reverses) would both be needed to properly assess whether this represents lasting progress.
Travel graphs 22 questions
QUESTION 1
3 marks
Criterion A
Medium
The travel graph shows a car's journey.
a.
Find the distance travelled and the time taken.
[1] b.
Calculate the car's average speed for the journey.
[2] Show complete worked solution
(a)
Distance $=80$ km, time $=2$ hours.
(b)
$$\text{speed} = \frac{80}{2} = 40 \text{ km/h}$$
QUESTION 2
5 marks
Criterion A
Medium
The travel graph shows a cyclist's journey, including a rest stop.
a.
Find the distance travelled in the first hour, and the cyclist's speed during that hour.
[2] b.
Describe what happens between 1 hour and 1.5 hours, using the shape of the graph.
[1] c.
Find the cyclist's speed during the final stage (1.5 h to 2.5 h), and compare it to the speed in the first stage.
[2] Show complete worked solution
(a)
Distance $=40$ km in 1 hour, so speed $=40$ km/h.
(b)
The graph is flat (horizontal) here — the cyclist has stopped, remaining at 40 km for 30 minutes.
(c)
Distance covered: $90-40=50$ km in $2.5-1.5=1$ hour, so speed $=50$ km/h — faster than the first stage's 40 km/h.
QUESTION 3
7 marks
Criterion B
Medium
Investigate what different sections of a travel (distance-time) graph tell us about a journey, using the full 5-stage journey shown.
a.
Identify which stage(s) of the journey represent the traveller being STOPPED (not moving), and justify your answer using the graph's shape.
[2] b.
Calculate the speed during Stage 5 (3h to 4.5h), and explain what the DOWNWARD direction of this section of the graph means about the direction of travel.
[3] c.
Compare the speeds of all 3 moving stages (Stage 1: 40 km/h, Stage 3: 50 km/h, Stage 5: 60 km/h) and state which was fastest, connecting this to the steepness of each section on the graph.
[2] Show complete worked solution
(a)
Stage 2 (1h to 1.5h) and Stage 4 (2.5h to 3h) are flat/horizontal sections — since distance is not changing over time in these intervals, the traveller must be stationary.
(b)
Distance covered: $90-0=90$ km over $4.5-3=1.5$ hours, so speed $=90\div1.5=60$ km/h. The downward slope means distance from the starting point is DECREASING — the traveller is returning back toward the start.
(c)
Stage 5 (60 km/h) was the fastest, followed by Stage 3 (50 km/h), then Stage 1 (40 km/h). This matches the graph — Stage 5 has the steepest line, confirming that steeper sections of a travel graph always represent faster speeds.
QUESTION 4
4 marks
Criterion C
Medium
A classmate looks at the downward-sloping Stage 5 of the journey graph above and says: 'the car is going backwards in time, which doesn't make sense.'
a.
Explain the classmate's misunderstanding — what does a downward slope on a DISTANCE-time graph actually represent?
[2] b.
Explain why a travel graph could never have a section that goes 'backwards' along the time axis (right to left), regardless of what the traveller is doing.
[2] Show complete worked solution
(a)
Time always moves forward (left to right) on this type of graph — a downward slope does NOT mean 'going backwards in time'. It means the DISTANCE from the starting point is decreasing as time moves forward, i.e. the traveller is heading back toward where they started.
(b)
Time is the independent variable on the horizontal axis, and it only ever increases — a traveller can change their DISTANCE (moving closer to or further from the start), but they can never make time itself run backwards, so the graph must always move rightward as time progresses, even while distance goes up or down.
QUESTION 5
6 marks
Criterion D
Medium
A delivery driver's morning route is shown in the travel graph.
a.
Find the driver's average speed for the ENTIRE journey (from start to the final point shown), including the rest stop time.
[3] b.
The driver's manager wants to know the speed while ACTUALLY DRIVING (excluding the stopped time). Find this, and explain why it's different from your answer in (a).
[3] Show complete worked solution
(a)
Total distance $=140$ km. Total time $=3.5$ hours. Average speed $=140\div3.5=40$ km/h.
(b)
Driving time only $=1.5+1.5=3$ hours (excluding the 0.5 hour stop). Speed while driving $=140\div3\approx46.7$ km/h. This is higher than the overall average (40 km/h) because the overall average also 'spreads' the stopped time across the whole journey, effectively counting it as if the driver was moving very slowly during that period too.
QUESTION 6
6 marks
Criterion A
Medium
The travel graph shows a delivery truck's journey, including a rest stop.
a.
Find the truck's speed during the FIRST stage (0 to 1.5 hours).
[2] b.
Find the truck's speed during the THIRD stage (2 to 3.5 hours).
[2] c.
Find the truck's OVERALL average speed for the entire journey (including the rest stop time).
[2] Show complete worked solution
(a)
$$60\div1.5=40\text{ km/h}$$
(b)
$$(150-60)\div1.5=60\text{ km/h}$$
(c)
$$150\div3.5\approx42.9\text{ km/h}$$
QUESTION 7
2 marks
Criterion A
Easy
The travel graph shows a train's journey between two cities, with a station stop.
a.
Find the total journey time and total distance.
[2] Show complete worked solution
(a)
Total time: 4 hours. Total distance: 220 km.
QUESTION 8
6 marks
Criterion B
Medium
Investigate which stage of the journey shown represents the FASTEST travel, and connect this to the visual STEEPNESS of the graph.
a.
Calculate the speed during Stage 1 (0-1h) and Stage 3 (1.5-2.5h).
[3] b.
Which stage is faster, and how does this show up VISUALLY in the graph (referring to the steepness/gradient of each line segment)?
[3] Show complete worked solution
(a)
Stage 1: $50\div1=50$km/h. Stage 3: $(130-50)\div1=80$km/h.
(b)
Stage 3 (80km/h) is faster than Stage 1 (50km/h). This is shown visually by Stage 3's line segment being STEEPER (rising more sharply per unit of time) than Stage 1's segment — a general rule for travel graphs is that greater steepness always corresponds to greater speed.
QUESTION 9
5 marks
Criterion B
Medium
Investigate what a LONG horizontal (flat) section of a travel graph — much longer than the moving sections — might suggest about a journey, using the graph shown.
a.
Compare the DURATION of the moving stage (0-1h) to the duration of the stopped stage (1h to 3h). Which is longer?
[2] b.
Suggest a plausible real-world scenario that would explain a stop lasting LONGER than the actual travel time, and discuss whether this journey pattern (short travel, long stop) seems unusual or could be entirely normal depending on context.
[3] Show complete worked solution
(a)
Moving stage: 1 hour. Stopped stage: 2 hours — the stop is TWICE as long as the travel itself.
(b)
A plausible scenario: a delivery driver travels a short distance to a client site (1 hour), then spends 2 hours actually completing a job/installation/meeting before returning. This pattern isn't unusual at all in many real contexts (e.g. service calls, business meetings, medical appointments) — a long 'stop' relative to short travel time often reflects the actual PURPOSE of the trip being time-consuming, with the travel itself being just a small part of the overall outing.
QUESTION 10
5 marks
Criterion B
Medium
A classmate looks at the graph shown (distance rising then falling back to 0) and says: 'the traveller went 80km then came back, so total distance travelled is 0km since they ended where they started.'
a.
Explain the classmate's error, distinguishing between DISPLACEMENT (net change in position, which could be 0) and TOTAL DISTANCE TRAVELLED (which accounts for the whole journey, including the return trip).
[3] b.
Calculate the correct total distance travelled for this journey.
[2] Show complete worked solution
(a)
The classmate is confusing 'ending back at the start' (which does mean the NET displacement is 0km) with 'total distance travelled' (which must count BOTH the outward AND return legs of the journey). Since the traveller went 80km out AND 80km back, the total distance travelled is $80+80=160$km, even though their final position is the same as where they started.
(b)
$$80+80=160\text{ km}$$
QUESTION 11
3 marks
Criterion A
Medium
A cyclist's journey is shown, with speeds of 60km/h in Stage 1 and Stage 3 (equal speeds, different stages).
a.
Verify that the speeds in Stage 1 (0-0.5h) and Stage 3 (1.5-2.5h) are indeed both 60km/h, showing your calculation for each.
[3] Show complete worked solution
(a)
Stage 1: $30\div0.5=60$km/h. Stage 3: $(90-30)\div1=60$km/h — confirmed both are 60km/h.
QUESTION 12
5 marks
Criterion B
Hard
Investigate whether TWO different stages of a travel graph can have the SAME speed but look VISUALLY DIFFERENT on the graph, using the cyclist graph above (Stage 1: 0.5h duration, Stage 3: 1h duration, both at 60km/h).
a.
Even though both stages have the SAME speed (60km/h), explain why their line segments on the graph do NOT look identical — what differs about them?
[2] b.
Explain the general principle: what property of a travel-graph line segment represents SPEED, and what property represents DURATION/DISTANCE covered — are these the same thing or different?
[3] Show complete worked solution
(a)
While both segments have the SAME STEEPNESS (gradient, since steepness represents speed), they differ in LENGTH along the graph — Stage 1 is a shorter segment (only 0.5 hours, covering 30km) while Stage 3 is a longer segment (1 hour, covering 60km), even though both segments rise at the exact same rate.
(b)
The STEEPNESS (gradient/slope) of a segment represents SPEED — two segments with the same steepness always represent the same speed, regardless of their length. The LENGTH of the segment (how far it extends along the graph) represents the DURATION and total DISTANCE covered during that stage — these are different properties: two segments can have identical steepness (same speed) while having very different lengths (different durations/distances), exactly as shown here.
QUESTION 13
4 marks
Criterion C
Medium
A student solving a travel graph problem writes only: '80/2=40'.
a.
Explain why this single unlabeled line, despite possibly being mathematically correct, is inadequate mathematical communication for a travel graph problem.
[2] b.
Rewrite this as a properly labelled solution, inventing a plausible travel graph context (e.g. a car travelling 80km in 2 hours during one stage of a journey).
[2] Show complete worked solution
(a)
Without any labels or context, a reader cannot tell WHAT the 80 and 2 represent (distance in km? time in hours? something else?), which STAGE of the journey is being analyzed, or what the resulting 40 actually MEANS (a speed? in what units?) — the calculation might be correct, but it communicates nothing meaningful on its own.
(b)
Example: 'During Stage 2 of the journey (from $t=1$h to $t=3$h), the car travelled from the 40km mark to the 120km mark — a distance of $120-40=80$km over $3-1=2$ hours. Speed during this stage: $80\div2=40$km/h.'
QUESTION 14
4 marks
Criterion C
Medium
A student calculates a NEGATIVE speed (e.g. '-60km/h') for a downward-sloping section of a travel graph, and is confused about what this means.
a.
Explain what a negative 'speed' actually represents in the context of a downward-sloping travel graph segment (referring to direction of travel, not an impossible physical speed).
[2] b.
State how a student should correctly communicate this in a written answer, distinguishing between 'speed' (always positive) and the signed 'rate of change of distance from start' (which can be negative).
[2] Show complete worked solution
(a)
A negative value here doesn't mean an impossible negative SPEED (speed itself — how fast something moves — is always positive or zero) — it reflects that DISTANCE FROM THE STARTING POINT is DECREASING, meaning the traveller is moving in the OPPOSITE direction (returning toward the start). The actual speed (magnitude, ignoring direction) would be the positive value $|{-60}|=60$km/h; the negative sign specifically indicates the direction of travel relative to the starting point, not the speed itself being negative.
(b)
A well-communicated answer should state: 'The rate of change of distance from the starting point is $-60$km/h, meaning the traveller is moving TOWARD the starting point at a speed of 60km/h' — explicitly separating the concept of speed (magnitude, always non-negative) from the signed rate that indicates direction.
QUESTION 15
8 marks
Criterion D
Hard
A logistics company plans delivery routes. The graph shows a driver's journey to a delivery point (90km away), followed by a REQUIRED 2-hour delivery/unloading stop (company policy), before the driver's shift ends (driver stays at the delivery point).
a.
Find the driver's speed during the travel stage, and the total time from departure to the end of the required stop.
[3] b.
The company wants to know if a driver could complete TWO such delivery trips (each with full 90km travel + 2hr stop, but with an IMMEDIATE 90km return leg at the same 60km/h speed after the stop) within a standard 8-hour shift. Calculate the total time for ONE complete round trip (there and back, with the stop), and determine how many such ROUND TRIPS fit within 8 hours.
[5] Show complete worked solution
(a)
Speed: $90\div1.5=60$km/h. Total time (travel + stop): $1.5+2=3.5$ hours.
(b)
One round trip: travel there ($1.5$h) + stop ($2$h) + travel back ($90\div60=1.5$h) $=1.5+2+1.5=5$ hours. Within an 8-hour shift: $8\div5=1.6$, so only 1 COMPLETE round trip fits (a second round trip would need 5 more hours, totalling 10 hours, exceeding the 8-hour shift).
QUESTION 16
2 marks
Criterion D
Easy
An airline's flight path is simplified to a single travel graph segment: a flight covering 2,400km in 3 hours (cruise speed only, ignoring takeoff/landing).
a.
Find the average cruise speed in km/h.
[2] Show complete worked solution
(a)
$$2400\div3=800\text{ km/h}$$
QUESTION 17
5 marks
Criterion D
Hard
A hiker's journey up a mountain trail is tracked, measuring cumulative trail distance covered.
a.
Assuming the vertical axis represents cumulative trail distance (km) and NOT elevation, find the hiker's pace (km/h) for the first stage.
[2] b.
Given your answer to (a) seems physically unrealistic for a hiker (80 km/h is faster than a car on many roads), identify what is likely WRONG with the problem's stated units or numbers, and suggest a more realistic total distance for a 2-hour hiking stage.
[3] Show complete worked solution
(a)
$$160\div2=80\text{ km/h — this is unrealistically fast for hiking, suggesting the units in this simplified model don't represent a literal hiking scenario.}$$
(b)
80km/h is wildly unrealistic for hiking (a fast human hiking pace is typically 3-6 km/h) — this suggests the graph's numbers (160 units in 2 hours) are almost certainly NOT meant to represent kilometres for a hiking context; a more realistic distance for a 2-hour hiking stage might be around 6-10km, meaning the graph's '160' value likely represents a different unit entirely (e.g. metres of elevation gain, not km of trail distance), and the problem as stated contains an inconsistency between its claimed units and realistic hiking speeds.
QUESTION 18
6 marks
Criterion B
Medium
Investigate whether a travel graph with only TWO moving stages (no stops) can still show a clear change in speed.
a.
Find the speed during Stage 1 (0-0.5h) and Stage 2 (0.5-1.5h).
[3] b.
Even though there's no flat (stopped) section anywhere in this graph, explain how you can still tell the traveller's speed CHANGED partway through the journey, just from the shape of the two connected segments.
[3] Show complete worked solution
(a)
Stage 1: $40\div0.5=80$km/h. Stage 2: $(100-40)\div1=60$km/h.
(b)
The two segments have DIFFERENT steepness (gradients) — Stage 1 is steeper (80km/h) than Stage 2 (60km/h) — this visible 'kink' or change in slope at the point where the segments meet ($t=0.5$h) directly shows the speed changed at that moment, even without any flat section indicating a stop; a travel graph doesn't need a stop to show varying speed, just a change in the line's steepness.
QUESTION 19
3 marks
Criterion C
Medium
A student writes a travel graph solution as: 'the graph shows the car went fast then slow', with no supporting calculations at all.
a.
Explain what specific pieces of quantitative evidence (numbers) this description is missing, and why 'fast' and 'slow' alone are not acceptable mathematical communication.
[3] Show complete worked solution
(a)
The description gives no actual SPEED VALUES (in km/h or similar units), no indication of WHEN the change from fast to slow occurred, and no distances or times to support the claim — 'fast' and 'slow' are subjective, relative terms with no fixed meaning, whereas proper mathematical communication requires specific, calculated, and clearly labelled quantities that another person could verify.
QUESTION 20
3 marks
Criterion C
Medium
Two students both correctly calculate a journey's average speed as 55km/h, but Student 1 shows full working with labelled steps, while Student 2 writes only the final answer '55km/h' with a small unlabeled calculation scribbled beside it.
a.
Even though both reach the correct answer, explain why Student 1's response would likely be assessed as demonstrating STRONGER mathematical communication, referencing what a reader can verify or understand from each response.
[3] Show complete worked solution
(a)
A reader examining Student 1's labelled, step-by-step work can VERIFY the reasoning is sound, follow exactly how the 55km/h was derived, and check for any errors in the process — with Student 2's response, a reader has no way to confirm whether the correct answer was reached through valid reasoning or through a lucky guess, memorized shortcut, or even a calculator error that happened to cancel out. Communication quality is judged by how well the REASONING is conveyed, not just whether the final number happens to be correct.
QUESTION 21
9 marks
Criterion D
Hard
A courier company evaluates a driver's route efficiency using the travel graph of a single delivery run.
a.
Find the speed during each of the two DRIVING stages (ignoring the brief stop).
[3] b.
The company's fuel-efficiency guideline recommends drivers maintain speeds under 75km/h whenever possible, since fuel efficiency drops significantly above this threshold. Identify which stage(s) VIOLATE this guideline, and estimate the ADDITIONAL fuel cost if that stage used 15% more fuel per km than a compliant stage would, given the violating stage covers 120km and fuel costs \$0.12 per km at the efficient rate.
[6] Show complete worked solution
(a)
Stage 1 (0-1h): $70\div1=70$km/h. Stage 2 (1.25-2.75h): $(190-70)\div1.5=80$km/h.
(b)
Stage 2 (80km/h) VIOLATES the 75km/h guideline; Stage 1 (70km/h) complies. Extra fuel cost for Stage 2: efficient cost would be $120\times0.12=\$14.40$; at 15% more, actual cost $\approx14.40\times1.15=\$16.56$. Additional cost due to speeding: $16.56-14.40=\$2.16$ for this one delivery run.
QUESTION 22
4 marks
Criterion D
Medium
A high-speed rail service covers 320km in 4 hours on a particular route.
a.
Find the train's average speed. A car covers the same distance at an average speed of $100\text{ km h}^{-1}$. Calculate each journey time, identify which journey is quicker and find the time difference.
[4] Show complete worked solution
(a)
The train's average speed is $$\frac{320}{4}=80\text{ km h}^{-1}.$$ The car's journey time is $$\frac{320}{100}=3.2\text{ h}.$$ The train takes $4.0$ h, so the car is quicker by $$4.0-3.2=0.8\text{ h}=\boxed{48\text{ min}}.$$