General Mathematics · Past QCAA questions All simple familiar questions
2021 · Paper 1 · Simple familiar

Question 24

[3 marks] Scientific calculator
Topic guidesPremiumTopic guide 4 · Reading a time series plot
Game plansPremiumGame plan 2 · Time series
The question

The population of fish in a fish farm pond was recorded every three months. The graph shows the data.

0200400600800100012001400 0481216 Quarters Population of fish

Describe the time series plot by identifying three key features.

Watch the situation first

Four quarters make a year, so the wiggles are a yearly cycle

Watch the plot get drawn, then read three things off it. The vertical axis starts at 400 here so the wiggles are easier to see.

One step along the axis is a quarter, three months Look past the wiggles and the population still climbs The peaks come round about every 4 quarters It bends up and down, so it is not a straight line Three features, one mark each 0481216 4006008001000 Quarters Population of fish 1 quarter = 3 months long-term trend, about +30 fish a quarter about 4 quartersabout 4 quartersabout 4 quarters 4 quarters × 3 months = 12 months Positive long-term trend Seasonal cycle every 12 months Non-linear form 3 features, 3 marks

Describe the shape of the whole plot, not single values. Then turn quarters into months before you name the cycle.

Now for the mathematics

Pick three statements and see which ones the guide pays for

Choose three statements to describe the plot. Each one you pick is drawn on the graph and gets its own feedback. Once you have three, the panel totals your marks against the 2021 marking guide.

0200400600800100012001400 0481216 Quarters Population of fish 1 quarter = 3 months long-term trend about 4 quartersabout 4 quartersabout 4 quarters fallsfallsfallsfalls 515968 highest, but on pattern
Your three features

Nothing chosen yet.

QCAA marking guide · 3 marks

Work through the solution one mark at a time

Try each step yourself before you reveal it. There is one mark for each feature, in the order the guide lists them.

Step 1 · form1 mark

Look at the overall shape. The plot climbs, dips, climbs again and dips again. It does not follow a straight line.

1. Non-linear form

That is the guide’s first feature. Be aware that this mark has dated. The 2025 syllabus keeps “form” for scatterplots and describes a time series by its trend, seasonality and irregular fluctuations. On a current paper, the feature to give alongside trend and seasonality is irregular fluctuations, and here they are small, with no sudden jumps or outliers.

Marker: correctly identifies the non-linear form

Step 2 · seasonality1 mark

The peaks sit near quarters 3, 7, 11 and 14, about 4 quarters apart. The stem says the fish were counted every three months, so each quarter is 3 months.

$$4\text{ quarters}\times3\text{ months}=12\text{ months}$$

2. Seasonal cycle every 12 months

The guide accepts “cycles every 12 months or equivalent”, so “every 4 quarters” or “once a year” earns it too. “Every 4 months” does not. That is the axis label read as months.

Marker: correctly identifies a seasonal pattern

Step 3 · trend1 mark

Ignore the yearly wiggle and compare like with like. Each peak is higher than the one before, and so is each dip. Over the four years the population grows by roughly 30 fish a quarter.

1. Non-linear form
2. Seasonal cycle every 12 months
3. Positive long-term trend

“Upward trend” is also accepted. Write “long-term” or “overall” so the marker can see you mean the trend and not every quarter, because the population falls after every peak.

Marker: correctly identifies a positive long-term trend

Putting it all together

Each phrase of the question gave you something

“recorded every three months”
One quarter is 3 months, so a cycle of 4 quarters is 12 months.
“The graph shows the data”
Everything comes from the shape of the curve. There is nothing to calculate.
“Describe the time series plot”
Describe the whole plot in words. Single readings such as 515 or 968 are not features.
“by identifying three key features”
Three different features. Two ways of saying the same seasonal pattern still count as one.
“[3 marks]”
One mark per feature. Number them so the marker can find all three.
What makes this simple familiar

The 2025 syllabus describes a time series by its trend, seasonality and irregular fluctuations, and keeps “form” for scatterplots, so the guide’s first mark is the one that has dated. The rest is standard Unit 3 work. The plot is clean, the stem tells you a quarter is three months, and the two features that still count today, trend and seasonality, are easy to see.

If I had to guess where you might lose marks, it would be the length of the cycle. The axis says “Quarters” and the peaks are 4 apart, so “every 4 months” is an easy slip to write. The other easy slip is “the population increases” with no “long-term”, when the plot clearly falls after every peak. The 2021 subject report has no comment on this question, so these are my guesses at the slips, not what markers reported.

Only mark this done when you could do it without help. Reading the solution does not count.

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Question wording and marking-guide steps are from the 2021 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2021, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.

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