Number questions hide negatives, multiples, remainders and counting inside a story; your child picks the operation, then checks the answer still makes sense. ACER's cooperative test for students entering Year 7 or 8 is on 27 February 2027.
A number question tests whole numbers, negatives, multiples and careful counting. In our ACER-style questions it doesn't look like a sum. It arrives as a story: a submarine diving, insects on long life cycles, a boat ferrying hikers.
The test is meant to measure mathematical ability rather than school achievement. In our number questions, the arithmetic is easy once your child sees it. Seeing it is the hard part.
Each number question goes the same way: read the situation, find the maths, then check it against the story.
Find the question word
Rise, next, smallest number of times: this word decides the operation, so find it before the numbers.
Turn the story into numbers
Below sea level becomes negative. A seat the ranger needs means a place fewer for hikers.
Choose what the situation needs
Meeting again means a common multiple, not a total. A gap between two depths means a difference.
Check the answer in the story
Is everyone on the far side? Does the submarine end at the turtle's depth? If not, look again.
Four worked examples
These are ours, written for the level of ACER's Level 1 Mathematics and ordered from easier to harder; none comes from an ACER paper. Each one shows the answer and the working; ask your child to name the operation as you read.
Worked explanation 1
A research submarine starts at the surface. It dives to 140 m below sea level, rises 35 m to film a reef, and then dives another 60 m. A turtle it has been filming is swimming at 80 m below sea level. How far must the submarine rise to reach the turtle's depth?
AA.
85 m
BB.
120 m
CC.
165 m
DD.
245 m
Answer: A
Answer: A (85 m)
Write depths below sea level as negatives. Start at −140 m, rise 35 m to −105 m, then dive 60 m to −165 m. The turtle is at −80 m. From −165 up to −80 is 165 − 80 = 85 m. The submarine must rise 85 m.
The trap: 245 m tempts children who add the two depths (165 + 80) instead of finding the gap between them; both are below the surface, so you subtract.
Why the other options are wrong
B (120 m): 120 m forgets the 35 m rise and measures from 200 m below up to 80 m.
C (165 m): 165 m is how deep the submarine is, not how far it must rise.
D (245 m): 245 m adds the two depths instead of finding the difference.
Worked explanation 2
In one forest, a kind of cicada comes above ground only once every 13 years. A second kind comes above ground only once every 17 years. Both kinds came above ground in 2015. In which year will both kinds next come above ground in the same year?
AA.
2045
BB.
2184
CC.
2236
DD.
2304
Answer: C
Answer: C (2236)
The first kind returns every 13 years after 2015 and the second every 17 years. They meet again after a number of years that is a multiple of both 13 and 17. 13 and 17 are both prime, so the smallest common multiple is 13 × 17 = 221. 2015 + 221 = 2236.
The trap: 2045 tempts children who add the two cycles (13 + 17 = 30). After 30 years neither kind is due: 30 is not a multiple of 13 or of 17.
Why the other options are wrong
A (2045): 2045 adds 13 and 17; 30 years is not a multiple of either cycle.
B (2184): 2184 is 2015 + 13 × 13, a multiple of 13 only.
D (2304): 2304 is 2015 + 17 × 17, a multiple of 17 only.
Worked explanation 3
A ranger must take 47 hikers across a river in a small boat. The boat holds 6 people, and the ranger must be in the boat to row it on every crossing. What is the smallest number of times the boat must cross the river, counting each trip in either direction, to get all the hikers to the far side?
AA.
10
BB.
15
CC.
19
DD.
20
Answer: C
Answer: C (19)
The ranger takes one of the 6 places, so each trip over carries 5 hikers. 47 hikers need 10 trips over (9 full loads of 5, then 2). Between trips over, the ranger must row back, which is 9 return trips. After the last load nobody needs to go back. 10 + 9 = 19 crossings.
The trap: 20 tempts children who give every trip over its own return trip, including the last one; once the last hikers are across, the boat stays there.
Why the other options are wrong
A (10): 10 counts only the trips over and forgets the boat has to come back.
B (15): 15 forgets the ranger takes a seat, so loads of 6 instead of 5.
D (20): 20 adds a return trip after the final load, which is not needed.
Worked explanation 4
A scoreboard shows each digit using lit bars. The table shows how many bars each digit uses: 0 uses 6, 1 uses 2, 2 uses 5, 3 uses 5, 4 uses 4, 5 uses 5, 6 uses 6, 7 uses 3, 8 uses 7 and 9 uses 6. How many two-digit numbers (from 10 to 99) use exactly 7 lit bars altogether?
Digit
Bars lit
0
6
1
2
2
5
3
5
4
4
5
5
6
6
7
3
8
7
9
6
AA.
3
BB.
6
CC.
7
DD.
8
Answer: D
Answer: D (8)
Every digit uses at least 2 bars, so split 7 into two parts of 2 or more: 2 + 5, 5 + 2, 3 + 4 or 4 + 3. 2 + 5: 1 then 2, 3 or 5 gives 12, 13, 15. 5 + 2: 2, 3 or 5 then 1 gives 21, 31, 51. 3 + 4 and 4 + 3 give 74 and 47. That is 3 + 3 + 1 + 1 = 8 numbers.
The trap: 6 tempts children who find the numbers using 1 with 2, 3 or 5, but forget the 3 + 4 split (7 and 4) altogether.
Why the other options are wrong
A (3): 3 counts only 12, 13 and 15, with 1 in the tens place.
B (6): 6 misses the pair made from the digits 7 and 4.
C (7): 7 finds 74 or 47 but not both orders.
What should your child do with a remainder?
School division can end with "remainder something". In a situation, the remainder has to be decided, not just written down.
Round up when everyone must be served. A boat load that is partly full is still a trip, so the leftover hikers need another crossing.
Round down when just the complete groups count, such as full boxes packed or whole teams formed.
Keep the remainder when the question asks what is left over.
Before dividing, your child should ask what the remainder would mean in the story. The answer tells them which way to go.
How can your child count without missing cases?
Some questions ask "how many ways" or "how many numbers". Guessing from a few examples can miss some.
The fix is a system. Split the problem into cases, list each case in order, then add the cases.
In the scoreboard question, that means listing every way to split the total number of bars between two digits, then checking each split in both orders. A child who lists in order can see when a case is missing; a child who lists at random can't.
What traps should your child watch for?
Each slip below turns the story into the wrong sum. When your child misses one, retell the story together and ask which word should have decided the operation.
Adding when the story needs a difference. Two depths below the surface are compared by subtracting, even though both sound like "going down".
Adding two cycles instead of finding a common multiple. After the sum of the two cycles, neither insect is due.
Forgetting a person takes up space. If the rower must be in the boat, it carries fewer passengers than its size.
Counting a return trip that isn't needed. Once the last group is across, the boat stays there.
Finding a pair but not its reverse. Two digits can sit in either order, and both orders count.
How fast does your child need to be?
The whole test has 32–36 multiple-choice questions in 40 minutes, so a number question can't take much longer than its share. Speed comes from fluency, not rushing. ACER does not allow rulers, books or papers, calculators, watches or mobile phones on test day, so times tables, mental subtraction and knowing the small prime numbers all pay off.
The scholarship tests are designed to be challenging, in ACER's words. A number question that takes a long read is normal; one that needs a long calculation probably has a shorter route your child hasn't spotted yet.
Turn everyday moments into small problems. Lift floors below ground are negative numbers.
Two buses on different timetables meet at a common multiple. Packing the car is a remainder question: how many trips for all the bags?
Ask your child to tell you the operation before doing any working, and why. Explaining the choice is the habit this whole topic depends on.
Four examples show the method. Practice test 1 has a full Mathematics section on ACER's time, and its results by skill show whether number questions are costing marks.
In the app, the number lesson comes as an Interactive version and as Notes, with skill drills on this topic. Number questions sit with the other kinds on the ACER Maths sample questions page. Patterns questions build on the same counting, and data questions use it to work out chance.
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.
