Patterns questions ask about a step too far ahead to draw; find the change and the start, then test the rule. For students entering Year 7 or 8, the cooperative test is on 27 February 2027.
A patterns question gives a few steps of something that grows or repeats, then asks about a step too far away to draw. It is early algebra: finding a rule and using it.
ACER's Mathematics test aims at mathematical ability rather than school achievement, so in our questions the pattern is never a plain list of numbers. It might be a chemist's chain of hexagons, a spring stretching under weights, or a weather station taking readings. Your child has to see the pattern inside the situation before they can continue it.
Read the situation, find the maths, check: for patterns, it looks like this.
Put the first few steps in a table
Step numbers down the side, the count or measurement beside them. A table shows the change at a glance.
Find the change and the start
How much does each step add? What would the value be at the step before the first?
Say the rule in words
Start with the first hexagon's corners, then add the same number for each extra hexagon. Words first; a formula later if it helps.
Test the rule on a step you can see
If the rule gives the right values for the steps the question shows, use it for the far step. If not, fix it now.
Four worked examples
We wrote these for the level of ACER's Level 1 Mathematics, easier first. They're our own, not ACER's. Each one shows the answer and the working, so draw the table with your child as you read.
Worked explanation 1
Chemists draw some molecules as a straight row of hexagons, where each new hexagon shares one side with the one before it. A row of 1 hexagon has 6 corners, a row of 2 hexagons has 10 corners and a row of 3 hexagons has 14 corners. How many corners does a row of 12 hexagons have?
AA.
50
BB.
54
CC.
60
DD.
72
Answer: A
Answer: A (50)
Each extra hexagon shares two corners with the one before, so it adds only 4 new corners. Starting from 6 corners for the first hexagon, the other 11 hexagons add 11 × 4 = 44. 6 + 44 = 50 corners.
The trap: 54 tempts children who start at 6 and add 4 for all 12 hexagons, counting the first hexagon twice.
Why the other options are wrong
B (54): 54 adds 4 corners for every hexagon, including the first, which already has 6.
C (60): 60 assumes each hexagon adds 5 new corners.
D (72): 72 is 12 × 6, ignoring the shared corners.
Worked explanation 2
In a science lesson, a class hangs masses on a spring and measures its length. With 50 g the spring is 13.5 cm long, with 150 g it is 16.5 cm long and with 250 g it is 19.5 cm long. If the pattern continues, what mass would make the spring exactly 30 cm long?
Mass hung (g)
Length of spring (cm)
50
13.5
150
16.5
250
19.5
AA.
350 g
BB.
550 g
CC.
600 g
DD.
1000 g
Answer: C
Answer: C (600 g)
Each extra 100 g stretches the spring 3 cm. From 250 g at 19.5 cm, the spring must grow another 30 − 19.5 = 10.5 cm. 10.5 ÷ 3 = 3.5 lots of 100 g, which is 350 g more. 250 + 350 = 600 g.
The trap: 1000 g tempts children who divide 30 cm by 3 cm, as if the spring had no length at all with no mass on it. With nothing hanging, it is already 12 cm long.
Why the other options are wrong
A (350 g): 350 g is only the extra mass needed beyond 250 g.
B (550 g): 550 g treats 13.5 cm as the length with no mass, but that length is for 50 g.
D (1000 g): 1000 g forgets the spring is already 12 cm long before any mass is added.
Worked explanation 3
An automatic weather station takes a reading every 7 minutes. Its first reading of the day is at 6:00 am. At what time does it take its 50th reading of the day?
AA.
11:36 am
BB.
11:43 am
CC.
11:50 am
DD.
11:57 am
Answer: B
Answer: B (11:43 am)
The 1st reading is at 6:00 am, the 2nd after one gap, the 3rd after two gaps. So the 50th reading comes after 49 gaps. 49 × 7 = 343 minutes, which is 5 hours 43 minutes. 6:00 am + 5 h 43 min = 11:43 am.
The trap: 11:50 am tempts children who multiply 50 × 7. The first reading happens with no wait, so there are only 49 gaps.
Why the other options are wrong
A (11:36 am): 11:36 am uses 48 gaps, one too few.
C (11:50 am): 11:50 am uses 50 gaps, but the first reading needs no gap.
D (11:57 am): 11:57 am uses 51 gaps, two too many.
Worked explanation 4
Tomasz builds a staircase shape from square tiles with sides of 2 cm. The top row has 1 tile, the next row has 2 tiles, the next has 3 tiles, and so on, with every row lined up on the left. The distance all the way around the outside of the shape is 64 cm. How many tiles has Tomasz used?
AA.
36
BB.
45
CC.
64
DD.
136
Answer: A
Answer: A (36)
Try small staircases: 1 row has a perimeter of 4 tile sides, 2 rows have 8, 3 rows have 12. The steps can be pushed out to make a square, so n rows always have 4n tile sides. Each tile side is 2 cm, so the perimeter is 8n cm. 8n = 64 gives n = 8 rows. Tiles: 1 + 2 + … + 8 = 36.
The trap: 136 tempts children who forget the tiles are 2 cm wide, treat 64 as 64 tile sides and get 16 rows (1 + 2 + … + 16 = 136).
Why the other options are wrong
B (45): 45 is the number of tiles for 9 rows, one row too many.
C (64): 64 mixes up the perimeter in centimetres with the number of tiles.
D (136): 136 forgets each tile side is 2 cm and finds 16 rows.
Why does the starting value matter so much?
The trap to watch for is the right change with the wrong start.
In the hexagon chain, the first hexagon has all its own corners, but every later one shares some with its neighbour. Treating the first like the others counts it twice.
In the spring question, the spring already has a length before anything hangs on it. Dividing the target length by the stretch per step ignores that length.
A quick test catches both: use the rule to work out a step the question already gives. If the rule misses, check the start first.
What is the fence-post trap?
A straight fence has a post at both ends, so it always has an extra post compared with the gaps. Readings, photos or bells at regular times work the same way: the first happens with no wait, so there is a gap before every event except the first.
The weather station question is built on this. Multiplying the number of the reading by the time between readings gives an answer that is a whole gap too late. Drawing the first few readings on a line shows the rule straight away.
Does your child need algebra with letters?
Not for our patterns questions. A rule in words, such as "start at this, add that each time", is enough for every question on this page.
If your child already knows a little algebra, they can write the rule as a formula with a letter for the step number. That's quicker for a far step, but it's a shortcut. What these questions reward is spotting the rule in an unfamiliar setting, and that comes before any notation.
What traps should your child watch for?
These are the slips the worked examples are built to catch. As you read the working, ask which wrong option looked right at first, then match it to a line below.
Counting the first step like the rest. The first shape or reading can behave differently.
Ignoring the value at the start. A spring, a candle or a tank may have something before the pattern begins.
Multiplying events instead of gaps. There is no gap before the first event, so count the gaps.
Forgetting the units in a shape pattern. A perimeter in centimetres is not a count of tile sides if each tile side is longer than a centimetre.
Guessing the rule from too little. A couple of values fit many rules; check another step before trusting it.
How long should a patterns question take?
The test has 32–36 multiple-choice questions in 40 minutes. A patterns question with a clear rule can be one of the quicker ones, once the table is drawn.
ACER does not allow rulers, books or papers, calculators, watches or mobile phones in the test, so the far step should be reachable by hand. If the working is getting very long, look for a rule your child hasn't used yet.
Build them. Matchsticks, blocks or coins make growing shapes, and your child can predict the count for a large step before checking. Ask "what does each new step add?" and "what was there at the start?" every time.
Fence posts are everywhere: stairs and landings, train stops, lamp posts along a street. Asking how many gaps there are between the first and the last makes the trap familiar before the test.
Four examples teach the habit. Practice test 1 has a full Mathematics section on ACER's time, and its results by skill show whether patterns are costing marks.
Our patterns lesson in the app has an Interactive version and Notes; skill drills then give more patterns questions. Patterns sit with the other kinds on the ACER Maths sample questions page. They draw on the counting in number questions, and many measurement questions are patterns in disguise.
See where your child stands
Practice test 1 marks every answer and shows results by skill: what your child already does well and the three skills to work on next.
Sample
Practise for the real test
Practice test 1 is free. Then packs from A$109 for 12 months, or A$9 a week, per exam.
