Worksheet generator

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 — 75KA-0023The multiplication principle

• AI added the number of options at each step instead of multiplying them.
• BI treated the two digits as one choice of ten rather than two independent choices of five.
• CI assumed the two digits had to be different, so I used 5 times 4 for them.
• EI used 5 choices for the letter position as well, forgetting there are only three letters.

4.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.

5.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.

6.B — 4KA-0069Inclusion–exclusion with two sets

• AI added 18 and 14 without removing the overlap, then subtracted from 30.
• CI subtracted the 6 who play both twice instead of once.
• DI subtracted only the footballers from the class and forgot the chess players entirely.
• EI found how many play at least one sport and gave that instead of how many play neither.

7.D — 24KA-0073Arrangements of a few objects

• AI counted the letters rather than their arrangements.
• BI multiplied 4 by 3 and stopped, forgetting the last two positions.
• CI used 4 choices for every position, as if letters could repeat.
• EI used 4 choices in each of four positions, which allows every letter to repeat.

8.B — 15KA-0074Selections of a few objects

• AI doubled the group size instead of counting pairs.
• CI allowed a student to be paired with themselves, adding six impossible pairs.
• DI counted each pair twice, once in each order, and forgot to halve.
• EI multiplied 6 by 6, counting every ordered pair including a student with themselves.

9.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.

10.A — 4KA-0154Inclusion–exclusion with two sets

• BI found how many students play at least one of the two and answered with that instead.
• CI added 18 and 15 to get 33, subtracted the 30 in the class, and used the 3 left over.
• DI removed the 7 from both groups and subtracted 11 and 8 from 30, taking the overlap away twice.
• EI answered with the number who play both, which is what the question gave me rather than what it asked for.

11.A — 56KA-0163Selections of a few objects

• BI counted ordered selections, so I counted the same three people once for every order they could stand in.
• CI multiplied the 8 members by the 3 places, which counts something quite different from a selection.
• DI divided the 336 ordered selections by 3 instead of by 3 factorial.
• EI chose 2 people rather than 3.

12.A — 60KA-0172Selections of a few objects

• BI chose any 4 from all 9 people and ignored the two-and-two requirement.
• CI added the two counts instead of multiplying them.
• DI treated the choices as ordered, counting the same committee several times.
• EI used 5 x 4 x 2 or a similar shortcut rather than counting each selection properly.

13.A — 6KA-0173Counting up to symmetry

• BI counted every colouring of the four edges and forgot that rotations make some of them identical.
• CI divided 16 by 4, but that only works when no painting is left unchanged by a rotation, and some are.
• DI halved 16, treating only the 180 degree turn as a symmetry.
• EI listed by how many edges are black but forgot that two black edges can be adjacent or opposite.