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Math Kangaroo practice

Name: ______________________Date: ____________4 questions · 12 points

1.The first three patterns are made of dots and follow a rule. How many dots will the fifth pattern have?[3]
Text description of the figure
Three dot patterns in a row. The first has one column of 2 dots with a single dot above it, making 3. The second has two columns of 2 dots with a single dot above, making 5. The third has three columns of 2 dots with a single dot above, making 7.
- A9
- B10
- C11
- D13
- E14
2.A pattern of tiles repeats: black, black, white. What colour is the 100th tile?[3]
- Ablack
- Bwhite
- Cit depends where you start counting
- Dgrey, because the pattern breaks
- Ethere is no 100th tile
3.A sequence begins 7, 11, 15, 19, and continues with the same constant step. What is the 30th term?[3]
- A123
- B127
- C120
- D119
- E137
4.A sequence is defined by a(1) = 2 and a(n+1) = 2 a(n) - 1 for every n. What is a(6)?[3]
- A33
- B31
- C64
- D17
- E63

Math Kangaroo Star · https://kangaroo-atlas.vercel.app · seed 1 · CC BY-NC-SA 4.0 · Independent project, not affiliated with Math Kangaroo in USA, NFP.

Answer key — Math Kangaroo practice

Each wrong choice carries the thinking that produces it. When a student picks C, this is what they were doing.

1.C — 11KA-0008Continuing a visual pattern
- AI continued the pattern to the fourth stage and stopped one stage early.
- BI counted only the columns of two and forgot the single dot sitting on top of each pattern.
- DI continued the pattern one stage too far and gave the sixth pattern instead of the fifth.
- EI doubled the third pattern's count, assuming the pattern doubles rather than grows by a fixed amount.
2.A — blackKA-0108Repeating cycles and position mod n
- BI got a remainder of 1 but read the last colour of the repeat rather than the first.
- CI thought the pattern had no fixed first tile, even though tile one is given.
- DI assumed a repeating pattern must eventually change.
- EI assumed the pattern runs only for as many tiles as one repeat contains.
3.A — 123KA-0160Arithmetic sequences and the nth term
- BI multiplied the step by 30 instead of by 29, forgetting that the first term needs no steps.
- CI used 4 times 30 and forgot to add the starting value.
- DI added 28 steps instead of 29, counting the gaps one short.
- EI treated the first term as the step and used 7 times 30 minus something close.
4.A — 33KA-0171Recursive rules
- BI used the rule 2a(n) - 1 but started the sequence at 1 instead of 2.
- CI doubled six times and ignored the minus one entirely.
- DI stopped at a(5), one term early.
- EI computed 2^6 - 1, applying the minus one only once at the very end.

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