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Math Kangaroo practice

Name: ______________________Date: ____________5 questions · 20 points

1.A large triangle is cut by two lines from its top corner into three small triangles side by side. How many triangles of any size appear in the picture?[4]
Text description of the figure
A large triangle with two straight lines drawn from its top vertex down to the base, dividing it into three small triangles side by side.
- A3
- B4
- C5
- D6
- E9
2.Small cubes are stacked in a corner as shown. Some are hidden from view. How many small cubes are there altogether?[4]
Text description of the figure
A stack of unit cubes in a corner, three layers tall. The bottom layer is a 3 by 3 square of 9 cubes, the middle layer is a 2 by 2 square of 4 cubes sitting on one corner of it, and the top layer is a single cube.
- A9
- B10
- C14
- D18
- E27
3.How many 2 by 2 squares fit on a 4 by 4 board, if they must line up with the grid?[4]
- A4
- B6
- C8
- D9
- E16
4.A coin is tossed three times. In how many of the possible outcomes are there exactly two heads?[4]
- A2
- B3
- C4
- D6
- E8
5.In how many ways can 7 be written as the sum of two different whole numbers bigger than zero? Swapping the order does not make a new way.[4]
- A3
- B4
- C6
- D7
- E12

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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.D — 6KA-0003Counting shapes hidden inside a figure
- AI counted only the three small triangles I could see as separate cells and stopped there.
- BI counted the three small triangles and added the whole big one, but forgot the two made of two cells.
- CI found the three small ones and one pair, but missed the second pair of neighbouring triangles.
- EI assumed every choice of two of the four lines would make a triangle, without checking each one.
2.C — 14KA-0009Counting cubes in a stack, including hidden ones
- AI counted only the cubes on the bottom layer and treated the rest as decoration.
- BI counted only the cubes whose faces I could actually see in the drawing.
- DI assumed every layer was a full 3 by 3 square and multiplied 9 by 2 for two layers.
- EI assumed the stack was a solid 3 by 3 by 3 block because it sits in a corner.
3.D — 9KA-0046Counting by position (sliding window)
- AI cut the board into four separate 2 by 2 blocks instead of sliding the square one step at a time.
- BI counted the three sliding positions along the top and doubled them for two rows.
- CI counted the positions along the top row and down the left column and added them.
- EI counted the sixteen small squares on the board rather than the 2 by 2 squares.
4.B — 3KA-0068Tree diagrams
- AI read the two in exactly two heads as the answer rather than as a condition to count.
- CI counted the outcomes with two or more heads, so I included the three-head one as well.
- DI counted the orderings of three different objects instead of listing the eight outcomes.
- EI gave the total number of possible outcomes rather than the number with exactly two heads.
5.A — 3KA-0076Casework
- BI included 0 and 7 as a pair, even though both numbers have to be bigger than zero.
- CI counted both orders of each pair, even though swapping does not make a new way.
- DI counted every starting number from 1 to 7 without checking which pairs repeat.
- EI counted both orders and also allowed pairs of equal numbers.

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